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MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 
Octavo,  Cloth,  $i.oo  each. 

No.  1.    HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 

No.  2.    SYNTHETIC  PROJECTIVE  GEOMETRY. 

By  George  Bruce  Halsted. 

N^No.  3.    DETERMINANTS. 

By  Laenas  Gifford  Weld. 

S^No.  4.    HYPERBOLIC   FUNCTIONS. 

By  James  McMahon. 

V^No.  5.    HARMONIC  FUNCTIONS. 

By  William  E.  Byerly. 

No.  6.    QRASSMANN'S  SPACE  ANALYSIS. 

By  Edward  W.  Hyde. 

No.  7.    PROBABILITY   AND  THEORY   OF    ERRORS. 

By  Robert  S.  Woodward, 

No.  8.    VECTOR  ANALYSIS  AND  QUATERNIONS, 

By  Alexander  Macfarlanr. 

No.  9.    DIFFERENTIAL  EQUATIONS. 

By  William  Woolsey  Johnson. 

Vno.  10.  THE  SOLUTION  OF  EQUATIONS. 

By  Mansfield  Merriman. 

No.  11.    FUNCTIONS  OF  A  COMPLEX  VARIABLE. 

By  Thomas  S.  Fiskb. 

PUBLISHED   BY 

JOHN  WILEY  &  SONS,   NEW  YORK. 

CHAPMAN  &  HALL,  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  3. 


DETERMINANTS. 


BY 

LAENAS   GIFFORD    WELD, 

Professor  of  Mathematics  in  the  State  University  of  Iowa. 


FOURTH    EDITION.  ENLARGED. 
FIRST   THOUSAND. 


NEW  YORK: 

JOHN   WILEY    &   SONS. 

London:    CHAPMAN  &  HALL,    Limited 

1906. 


Copyright,  1896, 

BY 

MANSFIELD  MERRIMAN  and  ROBERT  S.  WOODWARD 

UNDER   THE    TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906. 


ROBERT  DRUMMONP,  PRINTER,    NEW  YORK. 


EDITORS'  PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  ampHfy  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  Hterature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubHcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


235463 


AUTHOR'S   PREFACE. 


The  author  of  the  present  volume  feels  some  embarrassment 
in  having  already  offered  to  the  public  a  v^ork  upon  the  Theory 
of  Determinants.  The  apparently  general  acceptability  of  this 
former  v^ork,  which  has  now  reached  its  third  edition,  doubtless 
led  to  his  being  invited  by  the  editors  of  Higher  Mathematics 
to  prepare  for  them  a  chapter  upon  the  same  subject.  This 
was  done  without  the  least  thought  of  its  pubhcation  as  a  separate 
volume.  Now  that  its  issue  as  such,  along  with  the  other  chapters, 
is  requested  by  both  the  pubhshers  and  the  editors  of  Higher 
Mathematics,  it  is  but  just  to  the  author  that  the  above  circum- 
stances should  be  understood  lest  he  be  suspected  of  entertaining 
an  unseemly  desire  to  keep  himself  before  the  mathematical 
public  by  vain  repetition. 

The  limitations  imposed  have  permitted  the  addition  of  only 
a  few  articles  to  the  work  as  originally  pubHshed;  principally 
those  treating  of  linear  substitutions,  quantics,  invariance,  co- 
variance,  and  functional  determinants.  Determinants  of  special 
forms  have  not  been  considered,  nor  is  there  the  least  reference 
to  the  apphcation  of  determinants  to  geometry.  It  is  hoped, 
however,  that  the  work  may  prove  useful  to  the  constantly  increas- 
ing number  of  students  who,  while  not  wishing  to  specialize  in 
mathematics,  desire  to  obtain  the  comprehensive  view  of  its 
methods  and  processes  essential  to  the  successful  pursuit  of  the 
exact  sciences  in  general. 

Iowa  City,  Iowa,  U.  S.  A., 
December,  1905. 


CONTENTS. 


Art.  I.  Introduction Page  7 

2.  Permutations •.     .  8 

3.  Interchange  of  Two  Elements 9 

4.  Positive  and  Negative  Permutations 10 

5.  The  Determinant  Array 10 

6.  Determinant  as  Function  of  w^  Elements 11 

7.  Examples  of  Determinants 12 

8.  Notations 13 

9.  Second  and  Third  Orders 14 

10.  Interchange  of  Rows  and  Columns .16 

11.  Interchange  of  Two  Parallel  Lines 17 

12.  Two  Identical  Parallel  Lines 17 

13.  Multiplying  by  a  Factor 18 

14.  A  Line  of  Polynomial  Elements 18 

15.  Composition  of  Parallel  Lines 19 

16.  Binomial  Factors 20 

17.  Co-factors;  Minors 21 

18.  Development  in  Terms  of  Co-factors 23 

19.  The  Zero  Formulas 25 

20.  Cauchy's  Method  of  Development 26 

21.  Differentiation  of  Determinants 28 

22.  Raising  the  Order 29 

23.  Lowering  the  Order 30 

24.  Solution  of  Linear  Equations 31 

— '  25.  Consistence  of  Linear  Systems 32 

26.  The  Matrix 35 

27.  Homogeneous  Linear  Systems 35 

28.  Co-factors  in  a  Zero  Determinant ,     ...  37 

29.  Sylvester's  Method  of  Elimination 38 

30.  The  Multiplication  Theorem 40 

31.  Product  of  Two  Rectangular  Arrays 43 

32.  Reciprocal  Determinants 44 

33.  Linear  Transformations 44 

34.  Qu antics;  Invariants  and  Covariants 45 


'6  "  '     '  '    '     '  CONTENTS. 

Art.  35,  The  Discriminant ' Page  46 

36.  Composite  Quadrics 47 

37.  Discriminant  of  Binary  Qu antic  an  Invariant.     .     .     .     .  47 

38.  The  Jacobian .48 

39.  Jacobian  of  Indirect  Functions 49 

40.  The  Jacobian  a  Covariant ,  50 

41.  Jacobian  of  Implicit  Functions 50 

42.  The  Hessian 51 

Index 55 


DETERMINANTS. 


Art.  1.     Introduction. 

As  early  as  1693  Leibnitz  arrived  at  some  vague  notions 
regarding  the  functions  which  we  now  know  as  determinants. 
His  researches  in  this  subject,  the  first  account  of  which  is 
contained  in  his  correspondence  with  De  L'Hospital,  resulted 
simply  in  the  statement  of  some  rather  clumsy  rules  for  elimi- 
nating the  unknowns  from  systems  of  linear  equations,  and 
exerted  no  influence  whatever  upon  subsequent  investigations 
in  the  same  direction.  It  was  over  half  a  century  later,  in 
1750,  that  Gabriel  Cramer  first  formulated  an  intelligible  and 
general  definition  of  the  functions,  based  upon  the  recognition 
of  the  two  classes  of  permutations,  as  presently  to  be  set  forth. 

Though  Cramer  failed  to  recognize,  even  to  the  same  extent 
as  Leibnitz,  the  importance  of  the  functions  thus  defined,  the 
development  of  the  subject  from  this  time  on  has  been  almost 
continuous  and  often  rapid.  The  name  "  determinant"  is  due 
to  Gauss,  who,  with  Vandermonde,  Lagrange,  Cauchy,  Jacobi, 
and  others,  ra,nks  among  the  great  pioneers  in  this  development. 

Within  recent  years  the  theory  of  determinants  has  come 
into  very  general  use,  and  has,  in  the  hands  of  such  mathema- 
ticians as  Cayley  and  Sylvester,  led  to  results  of  the  greatest 
interest  and  importance,  both  through  the  study  of  special 
forms  of  the  functions  themselves  and  through  their  applica- 
tions.* 

*  A  list  of  writings  on  Determinants  is  given  by  Muir  in  Quarterly  Journal 
of  Mathematics,  1881,  Vol.  XVIII,  pp.  1 10-149. 


DETERxMINANTS. 


Art.  2.    Permutations. 


The  various  orders  in  which  the  elements  of  a  group  may 
be  arranged  in  a  row  are  called  their  permutations. 

Any  two  elements,  as  a  and  b,  may  be  arranged  in  two 
orders :  ab  and  ba.  A  third,  as  ^,  may  be  introduced  into  each 
of  these  two  permutations  in  three  ways  :  before  either  element, 
or  after  both  ;  thus  giving  3x2  =  6  permutations  of  the  three 
elements.  In  like  manner  an  additional  element  may  be  intro- 
duced into  each  of  the  permutations  of  i  elements  in  (/+  i) 
ways:  before  any  one  of  them,  or  after  all.  Hence,  in 
.general,  if  Pi  denote  the  number  of  permutations  of  i  ele- 
ments, Pi^^  =  {t+  i)Pi.  Now,  P3  =  3  X  2  X  I  =  3  ! ;  hence 
P^  =  4  X  3^=41;  and,  n  being  any  integer, 

P„  =  n{n  —  i){n  —  2) . .  .  i  =  n\. 
That  is,  the  number  of  permutations  of  n  elements  is  n\. 

For  all  integral  values  of  n  greater  than  unity,  n !  is  an 
•even  number. 

If  the  elements  of  any  group  be  represented  by  the  differ- 
ent letters,  a,  b,  c, . .  .,  the  alphabetical  order  will  be  considered 
as  the  natural  order  of  the  elements.  If  represented  by  the 
same  letter  with  different  indices,  thus : 

a^,  ^„  ^3, . . . ;     or  thus :     a\  a" ^  a'" , . . ., 

the  natural  order  of  the  elements  is  that  in  which  the  indices 
form  a  continually  increasing  series. 

Any  two  elements,  whether  adjacent  or  not,  standing  in 
their  natural  order  in  a  permutation  constitute  a  permanence ; 
standing  in  an  order  which  is  the  reverse  of  the  natural,  an 
inversion.  Thus,  in  the  permutation  daecb^  the  permanences 
are  de,  ae,  ab,  ac ;  the  inversions,  da,  dc,  db,  ec,  eb,  cb. 

The  permutations  of  the  elements  of  a  group  are  divided 
into  two  classes,  viz.:  even  or  positive  permutations,  in  which 
the  number  of  inversions  is  even  ;  and  odd  or  negative  permu- 
tations, in  which  the  number  of  inversions  is  odd. 

4i. 


INTERCHANGE    OF    TWO    ELEMENTS.  9 

When  the  elements  are  arranged  in  the  natural  order  the 
number  of  inversions  is  zero — an  even  number. 

Thus,  the  even  or  positive  permutations  of  the  elements 
Y7,,  ^„  ^3  are 

while  the  odd  or  negative  permutations  are 

Art.  3.    Interchange  of  Two  Elements. 

It  will  now  be  shown  that  if,  in  any  permutation  of  the 
elements  of  a  group,  two  of  the  elements  be  interchanged  the 
class  of  the  permutation  will  be  changed. 

Let  q  and  j  be  the  elements  in  question.  Then,  represent- 
ing collectively  all  the  elements  which  precede  these  two  by 
P,  those  which  fall  between  them  by  R,  and  those  which  follow 
by  Ty  any  permutation  of  the  group  may  be  written 

PqRsT. 
Of  the  elements  R,  supposed  to  be  r  in  number,  let  represent 
h  the  number  of  an  order  higher  than  q, 
i    "         "  "    ''        "      lower      "     q, 

j    "         **  "    "       "      lower      "     s, 

k     "         "  "    "        "      higher     "     s. 

It  is  evident  that  no  change  in  the  order  of  the  elements  qRs 

■can  affect  their  relations  to  the  elements   of   either  P  or   T. 

Then,  passing  from  the  order  PqRsT  to  the  order 

PRqsT 

changes  the  number  of  inversions  by  {h  —  i) ;  and  passing  from 

this  to  the  order 

PsRqT 

again  changes  the  number  of  inversions  by  (/  —  /&)  ±  i,  the 

*    p!"^    [sign  being  used  as  q  is    of  |  ^9^^^^  I  order   than    s. 
\  mmus  ^     ^  ^  ^  (  higher  j 

The  total  change  in  the  number  of  inversions  due  to  the  inter- 
change of  the  two  elements  in  question  is,  therefore, 

h  —  i  +y  ^  k  ±  \. 


10  ..  DETERMINANTS. 

But  since  i  =^  r  —  h  and  k  ^=  r  —j\  this  may  be  written 

2{h+J-r)±  I, 
which  is  an  odd  number  for  all  admissible  values  of  //,/,  and  r^ 
Hence,  the  interchange  of  any  two  elements  in  a  permutation 
changes  the   number  of   inversions  by  an   odd  number,  thus 
changing  the  class  of  the  permutation. 

Art.  4.    Positive  and  Negative  Permutations. 

Of  all  the  permutations  of  the  elements  of  a  group,  one 
half  are  even  and  one  half  odd. 

To  prove  this,  write  out  all  the  permutations.  Now  choose 
any  two  of  the  elements  and  interchange  them  in  each  permu- 

Itation.     The  result  will  be  the   same  set  of   permutations  as 

{even  ) 
odd   \  P^^""*uta- 

tion  of  the  old  set  has  been  converted  into  an  \  p„p,,  r  one  in 
the  new.     Hence,  in  either  set,  there  are  as  many  even  permu- 
tations as  odd  ;  that  is,  one  half  are  even  and  one  half  odd. 
Prob.  I.  Classify  the  following  permutations: 
{\)  b  c  d  e  a  \  (2)   iii  v  i  ii  iv;  (3)  kninilj', 

(4)  a"  a'a'  a^'  a'";  (5)  l^eyt^ad-,  (6)   52413; 

(7)  -^1 -^3 -^0 -^4  ^a -^5 ;  (S)  F.  Tu.  M.  Th.  W.;  (9)  jxkviX, 
Prob.  2.   Derive  the  formula  for  the  number  of  permutations  of 
n  elements  taken  m  at  a  time.     (Ans.  n\/{7t  —  m)\.) 

Prob.  3.  How  many  combinations  of  m  elements  arranged  in  the 
natural  order  may  be  selected  from  a  group  of  n  elements?  (Ans. 
n\/m\(n  —  m)\.) 

Prob.  4.  Show  that  o!  =  i. 

Art.  5.    The  Determinant  Array. 

Assume  ft^  elements  arranged  in  n  vertical  ranks  or  columns, 
and  n  horizontal  ranks  or  rows,  thus : 


a„  a^    . . .  a 


(«) 


DETERMINANT  AS  FUNCTION  OF  «'  ELEMENTS.        11 

In  this  array  all  the  elements  in  the  same  column  have  the 
same  superscript,  and  those  in  the  same  row  the  same  subscript. 
The  columns  being  arranged  in  order  from  left  to  right,  and 
the  rows  likewise  in  order  from  the  top  row  downward,  the 
position  of  any  element  of  the  array  is  shown  at  once  by  its 
indices.  Thus,  a^"  is  in  the  third  column  and  the  fifth  row 
of  the  above  array. 

The  diagonal  passing  through  the  elements  a^ ^  ^/', . .  .  ^^^"^ 
is  called  the  principal  diagonal  of  the  array;  that  passing 
through  a^^  cin-i" y  •  •  •  '^/"^  the  secondary  diagonal.  The  posi- 
tion occupied  by  the  element  a/  is  designated  as  the  leading 
position. 


Art.  6.    Determinant  as  Function  of  «'  Elements.    ■^■ 

The  array  just  considered,  inclosed  between  two  vertical 
bars,  thus  : 

a/  a/'  .  .  .  ^/«> 


a„  a„    ...  a„' 

is  used  in  analysis  to  represent  a  certain  function  of  its  «'  ele- 
ments called  their  determinant.*  This  function  may  be  defined 
as  follows : 

Write  down  the  product  of  the  elements  on  the  principal 
diagonal,  taking  them  in  the  natural  order ;  thus  : 

This  product  is  called  the  principal  term  of  the  determinant. 
Now  permute  the  subscripts  in  this  principal  term  in  every 
possible  way,  leaving  the  superscripts  undisturbed.  To  such  of 
the  n !  resulting  terms  as  involve  the  even  permutations  of  the 
subscripts  give  the  positive  sign ;  to  those  involving  the  odd 

*  This   notation    was   first   employed   by  Cauchy    in    1815.      See    Dostpr's 
Th6orie  des  determinants,  Paris,  1877. 


12 


DETERMINANTS. 


permutations,  the  negative  sign.  The  algebraic  sum  of  all  the 
terms  thus  formed  is  the  determinant  represented  by  the 
given  array. 


Art.  7.    Examples  of  Determinants. 

Applying  the  process  above  explained  to  the  array  of  four 
elements  gives 

a;  a^'    ~a;a^'  -  a^a^'.  (i> 


As  an  example  of  a  determinant  of  nine  elements,  with  its  ex- 
pansion, may  be  written 


<  a:'  a:" 


-  a;a;'a;"  -  a: a:' a;"  -  a^a^'a,'".     (2) 


It  is  evident,  from  the  mode  of  its  formation,  that  each  term' 
of  the  expansion  of  a  determinant  contains  one,  and  only  one, 
element  from  each  column  and  each  row  of  the  array. 

It  follows  that  every  complete  determinant  is  a  homoge- 
neous function  of  its  elements.  The  degree  of  this  function, 
with  respect  to  its  elements,  is  called  the  order  of  the  deter- 
minant. Thus,  (i)  and  (2)  are  of  the  second  and  third  order 
respectively. 

The  definition  of  a  determinant  given  in  the  preceding 
article  is  once  more  illustrated  by  the  following  example  of  a 
determinant  of  the  fourth  order  with  its  complete  development : 


^1  ^1  ^1  ^1 
a,  b,  c^  d. 


^  +  <^^b^c^d^  —  afi^c^d^  —  ajb^c^d^  +  afi^c^d^  ^ 
+  afi^c^d^  —  ap^c^d^  —  aj)^c^d^  +  ajb^c^d^ 

-\-  ajb^c^d^  —  aj?^c^d^  —  ajb^c^d^  -\-  aj).^c^d^ 
-\-  aj)^c^d^  —  ajb^c^d^  —  aj)^c^d^  +  ^■pK'^^d^ 
-\-  ajb^c^d^  —  ajb^c^d^  —  ajj^c^d^  -\-  afi^c^d^ 


(3) 


NOTATIONS. 


la 


It  will  be  noticed  that,  in  this  case,  the  columns  are  ranked 
alphabetically  instead  of  by  the  numerical  values  of  a  series  of 
indices. 


Art.  8.    Notations. 

Besides   the  notations  already  employed,  the  following  is. 
very  extensively  used  : 


.  .  a^ 


This  is  called  the  double-subscript  notation  ;  the  first  subscript 
indicating  the  rank  of  the  row,  the  second  that  of  the  column. 
Thus  the  element  ^3:53  is  in  the  second  row  and  the  third  column. 
The  letters  are  sometimes  omitted,  the  elements  being  thus 
represented  by  the  double  subscripts  alone.* 

Instead  of  writing  out  the  array  in  full,  it  is  customary, 
when  the  elements  are  merely  symbolic,  to  write  only  the  prin- 
cipal term  and  enclose  it  between  vertical  bars.  This  is  called 
the  umbral  notation.  Thus,  the  determinant  of  the  «th  order 
is  written 


(«) 


or,  using  double  subscripts. 


>    ^UH     I 


These  last  two  forms  are  sometimes  still  further  abridged  to 


(«) 


and 


'^pectively. 


Prob.  5.  Write  out  the  developments  of  the  following  determi- 

11  ants: 

(i) 


a,b, 

;     {2) 

P'P" 

;     (3) 

p>    g' 

;     (4) 

a  b 

"^K 

/?" 

P"  g" 

a/? 

*  Leibnitz  indicated  the  elements  of  a  determinant  in  this  same  manner,, 
though  he  made  no  use  of  the  array. 


li 

DETERMINANTS. 

(S)  1  '^A^,  1 

;   (6) 

P'P"P"' 

r'/'  r'" 

;   (7) 

/'   q'   r' 
p"  q"  r" 
p"'q"'r"' 

■;    (8) 

{9)   I    II»   22 


a  b  c 
a  Py 

X  y  z 

Jo)  I  ^1,3  h     (11)  I  ^o'^^«.  I  ;     (12)  I  a,,a,,a,,a,. 


Prob.  6.  How  many  terms  are  there  in  the  development  of  the 
determinant  |  a^^  \  ? 

In  the  above  determinant  tell  the  signs  of  the  terms  : 

(3)  a:a:'ar'xr«'. 

Prob.  7.  Show  that  in  the  expansion  of  any  determinant,  all  of 
whose  elements  are  positive,  one  half  the  terms  are  positive  and  one 
half  negative. 

Prob.  8.  In  determinants  of  what  orders  is  the  term  containing 
the  elements  on  the  secondary  diagonal  (called  the  secondary  term) 
positive  ? 

Prob.  9.  What  is  the  order  of  the  determinant  whose  secondary 
term  contains  10  inversions  ?  36  inversions  ? 

Prob.  10.  In, the  expansion  of  a  determinant  of  the  «th  order, 
how  many  terms  contain  the  leading  element  ? 

Art.  9.    Second  and  Third  Orders. 


Simple  rules  will  now  be  given  for  writing  out  the  expan- 
sions of  determinants  of  the  second  and  third  orders  directly 
from  the  arrays  by  which  they  are  represented. 

To  expand  a  determinant  of  the  second  order,  write  the 
product  of  the  elements  on  the  principal  diagonal  minus  the 
product  of  those  on  the  secondary  diagonal,  thus : 

=  ad  —  be. 

=  -3  +  1 

The  following  method  is  applicable  to  determinants  of  the 
third  order  :* 

*  This  method  was  first  given  by  Sarrus,  and  is  often  called  the  rule  of  Sarrus; 
sec  Finck's  6l6ments  d'Algebre,  1846,  p.  95. 


Likewise, 


a     b 

c     d 

-9     5 

2     il 

=  -  3  + 10  =  7. 


SECOND    AND    THIRD    ORDERS. 


15 


Beneath  the  square  array  let  the  first  two  rows  be  repeated  in 
order,  as  shown  in  the  figure. 
Now  write  down  six  terms,  each 
the  product  of  the  three  ele- 
ments lying  along  one  of  the 
six  oblique  lines  parallel  to  the 
diagonals  of  the  original  square. 
Give  to  those  terms  whose  ele- 
ments lie  on  lines  parallel   to  __. 
the  principal  diagonal  the  posi- 
tive  sign ;   to  the   others,  the  - 
negative  sign.     The   result   is 
the  required  expansion.     Ap- 
plying the  method  to  the  determinant  just  written  gives 

After  a  little  practice  the  repetition  of  the  first  two  rows  will 
be  dispensed  with. 

The  above  methods  are  especially  useful  in  expanding 
determinants  whose  elements  are  not  marked  with  indices,  or 
in  evaluating  those  having  numerical  elements.  No  such  sim- 
ple methods  can  be  given  for  developing  determinants  of  higher 
orders,  but  it  will  be  shown  later  that  these  can  always  be 
resolved  into  determinants  of  the  third  or  second  order. 


Prob.  II.  Develop  the  following  determinants: 


(i) 


(4) 


(7) 


a  k  g  ; 
h  b  f 
g  f  c 

X,  y,  I 

I        cos  OL 
cos  Oi      I 


(^) 


(5) 


(8) 


0  —n  —m 

» 

(3) 

A   c 

b 

> 

n      0  —  / 

c    Ba 

m      I      0 

b    a  C 

I      P        Q 

;      (6) 

cos  a  sin  /? 

0  cos  a  sin  /? 

sin  a  cos  ^ 

0  sin  a  cos  ^ 

(9) 

1  v_  I 

> 

a  b  c 
cab 
b  c  a 

• 

4    V-2 

(l) 


Prob.  12.  Evaluate  the  following: 


1  2  3 
3  I   2 

2  3  I 


(2) 


-  2 

o 

12 


2  \ 

2   o 

2     I 


(3) 


—   I 

4/"^=~i 


-|/' 


V- 


(Ans, 


—  I 

;;  16;  2.) 


16  DETERMINANTS, 

Art.  10.    Interchange  of  Rows  and  Columns. 

Any  term  in  the  development  of  the  determinant  |  ^/"^j  may- 
be written 

±a^  a!'  a!"  .,.a^, 

in  which  ^2/'. .  ./is  some  permutation  of  the  subscripts  i,  2,  3,. . .«. 
Designate  by  u  the  number  of  inversions  in  hij .  .  .  /.  Also,  let 
V  be  the  number  of  interchanges  of  two  elements  necessary  to 
bring  the  given  term  into  the  form 

±  «/^)  a^'f^  a,^"-^ .  .  .  a^^'\ 
in  which  the  subscripts  are  arranged  in  the  natural  order,  while 
pqr . . .  /  is  a  certain  permutation  of  the  superscripts  ',  ", '" , . . .  ^^'>, 

This  permutation  is  even  or  odd  according  as  v  is  even  or 
odd.  But  u  and  v  are  obviously  of  the  same  class ;  that  is, 
both  are  even  or  both  odd.  Hence  the  permutations  hij . .  .  / 
and  pqr . . ,  /  are  of  the  same  class ;  and  the  term  will  have  the 
same  sign,  whether  the  sign  be  determined  by  the  class  of  the 
permutation  of.  the  subscripts  when  the  superscripts  stand  in 
the  natural  order,  or  by  the  class  of  the  permutation  of  the 
superscripts  when  the  order  of  the  subscripts  is  natural. 

It  follows  that  the  same  development  of  the  determinant 
array  will  be  obtained  if,  instead  of  proceeding  as  indicated  in 
Art.  6,  the  superscripts  of  the  principal  term  be  permuted,  the 
subscripts  being  left  in  the  natural  order,  and  the  sign  of  each 
of  the  resulting  terms  written  in  accordance  with  the  class  of 
the  permutations  of  its  superscripts. 

Passing  from  one  of  these  methods  of  development  to  the 
other  amounts  to  the  same  thing  as  changing  each  column  of 
the  array  into  a  row  of  the  same  rank,  and  vice  versa.  Hence, 
a  determinant  is  not  altered  by  changing  the  columns  into  cor- 
responding rows   and  the   rows  into  corresponding   columns. 

Thus : 

a,     a^     .  .  .  a^ 


a,'  a,''  .  .  .  ^Z'') 


a,   a. 


(n) 


a'  a"         ^  (") 


a. 


(«)  n  i*')  n  C') 


ar'  a. 


TWO    IDENTICAL    PARALLEL    LINES.  IT 

Whatever  theorem,  therefore,  is  demonstrated  with  reference 
to  the  rows  of  a  determinant  is  also  true  with  reference  to  the 
columns. 

The  rows  and  columns  of  a  determinant  array  are  alike 
called  lines. 

Art.  11.     Interchange  of  Two  Parallel  Lines. 

If  any  two  parallel  lines  of  a  determinant  be  interchanged, 
the  determinant  will  be  changed  only  in  sign. 

For,  interchanging  any  two  parallel  lines  of  a  determinant 
array  amounts  to  the  same  thing  as  interchanging,  in  every 
term  of  the  expansion,  the  indices  which  correspond  to  these 
lines.  Since  this  changes  the  class  of  each  permutation  of  the 
indices  in  question  from  odd  to  even  or  from  even  to  odd,  it 
changes  the  sign  of  each  term  of  the  expansion,  and  therefore 
that  of  the  whole  determinant. 

It  follows  from  the  above  that  if  any  line  of  a  determinant 
be  passed  over  in  parallel  lines  to  a  new  position  in  the  array 
the  new  determinant  will  be  equal  to  the  original  one  multi- 
plied by  (—  i)"'. 

The  element  a^^'^  may  be  brought  to  the  leading  position 
by  passing  the  ^th  row  over  the  {k  —  \)  preceding  rows,  and 
the  ^th  column  over  the  {s —  i)  preceding  columns.  This 
being  done  the  determinant  is  multiplied  by 

(-  1)*-  .  (-  ly-  =  (-  ir% 

which  changes  its  sign  or  not  according  as  {k-\-s)  is  odd  or 
even. 

The  position  occupied  by  a^'''  is  called  a  positive  position 
when  {k  +  s)  is  even  ;  a  negative  position  when  {k  -j-  s)  is  odd. 

Art.  12.    Two  Identical  Parallel  Lines. 

A  determinant  in  which  any  two  parallel  lines  are  identical 
is  equal  to  zero. 

For  the  interchange  of   these  two  parallel  lines,  while  it 


18 


DETERMINANTS. 


changes  the  sign  of  the  determinant,  will  in  no  way  alter  its 
value.     The  value  then,  if  finite,  can  only  be  zero. 

Art.  13.    Multiplying  by  a  Factor. 

Multiplying  each  element  of  a  line  of  a  determinant  by  a 
given  factor  multiplies  the  determinant  by  that  factor. 

Since  each  term  of  the  development  contains  one  and  only 
one  element  from  the  line  in  question  (Art.  7),  then  multiply- 
ing each  element  of  this  line  by  the  given  factor  multiplies 
each  term  of  the  development,  and  therefore  the  whole  deter- 
minant, by  the  same  factor. 

It  follows  that,  if  the  elements  of  any  line  of  a  determinant 
contain  a  common  factor,  this  factor  maybe  canceled  and  written 
outside  the  array  as  a  factor  of  the  whole  determinant ;  thus  : 


in  a. 


=  m 


A  determinant  in  which  the  elements  of  any  line  have  a 
common  ratio  to  the  corresponding  elements  of  any  parallel 
line  is  equal  to  zero.  For  this  common  ratio  may  be  written 
outside  the  array,  which  will  then  have  two  identical  lines.  Its 
value  is  therefore  zero  (Art.  12). 

A  determinant  having  a  line  of  zeros  is  equal  to  zero. 

Art.  14.    A  Line  of  Polynomial  Elements. 

A  determinant  having  a  line  of  elements  each  of  which  is 
the  sum  of  two  or  more  quantities  can  be  expressed  as  the 
sum  of  two  or  more  determinants. 


Let 


a.     (^,-^/  +  ^/'±...)     ^. 

^3  (^3-^/    +   ^3"±...)  ^3 


(I) 


be  such  a  determinant.     Then,  if 


COMPOSITION    OF    PARALLEL    LINES.  19" 

any  term  of  the  expansion  of  the  determinant  A  is 

±  Uh  BiCj  .  .  .  =  ±a^  diCy.  ,  .  ^  U),  bl  Cj . .  . 

±  aj,  bl'  ^y .  .  .  ±  .  .  .  (2) 

The  terms  in  the  expansion  of  A  are  obtained  by  permuting 
the  subscripts  h,  i,j\ ...  of  a^  Bi  Cj  .  ,  ,  ,  But  permuting  at 
the  same  time  the  subscripts  of  the  terms  in  the  second  mem- 
ber of  (2),  and  giving  to  each  term  thus  obtained  its  proper 
sign,  there  results 

which  proves  the  theorem. 

Art.  15.    Composition  of  Parallel  Lines. 

If  each  element  of  a  line  of  a  determinant  be  multiplied  by 
a  given  factor  and  the  product  added  to  the  corresponding  ele- 
ment of  any  parallel  line,  the  value  of  the  determinant  will  not 
be  changed  ;  thus: 


'1« 


This  will  appear  upon  resolving  the  second  member  into 
two  determinants  (Art.  14),  one  of  which  will  be  the  given  de- 
terminant, while  the  other,  upon  removal  of  the  given  factor, 
will  vanish  because  of  having  two  identical  lines. 

In  like  manner  any  number  of  parallel  lines  may  be  com- 
bined without  changing  the  value  of  the  determinant,  care 
being  taken  not  to  modify  in  any  way  the  elements  to  which 
are  added  multiples  of  corresponding  elements  from  other 
parallel  lines.     For  example,  |  ^,^„  |  is  equivalent  to 


20 


DETERMINANTS. 

Art.  16.    Binomial  Factors. 


A  determinant  which  is  a  rational  integral  function  of  a 
and  of  by  such  that  if  b  is  substituted  for  a  the  determinant 
vanishes,  contains  {a  —  ^)  as  a  factor.     For  example, 

A^  a^  —  p"^   a  —  q     a-\-r 
b'-p'    b-q     b  +  r 
p  q  r 

as  divisible  by  {a  —  b). 

To  prove  this,  let  the  expansion  of  any  such  determinant 
be  written  in  the  form 

the  coefficients  m^,  m^,  m,,  .  .  .  being  independent  of  a.      Now 
when  b  is  substituted  for  a  the  determinant  vanishes.     Hence, 

o  =  ;//o  +  m^b  -\-  mj}^  +  •  •  • 

Subtracting  this  from  the  preceding  gives 

A  =  m^{a  —  ^)  +  m^a^  —  3')  +  .  .  . 

This  being  divisible  by  {a  —  b),  the  theorem  is  proven. 

Prob.  13.  Prove  the  following  without  expansion  : 


(i) 


=  o; 


(3) 


(4) 


O      —X 

my      o 

—  mnz  nz 

b  -\-  c      a         a 

b      c  -{-  a     b 
c  c       a  -\-b 

b'  -^c" 


(2) 


=  2 


a 


b 
c 


«'  +  ^' 


o 
c 

b 

=    2 


o 
—  c 
b  —a 

c  b 
o  a 
a     o 


c  -  b 
o       a 


=  o; 


(5) 


a  s'mA 
b  sinB 
c    sin  C 


b  -  c 
c  —  a 
a  —  b 


o,  the  elements  referring  to   the 
triangle  ABC. 


CO-FACTORS  :     MINORS. 


21 


Prob.  14.  Prove  that 

I  X  —a  y 
IX -a  y, 
IX,- a  y. 


b 

:= 

1  X  y 

= 

b 

I  Xj  y^ 

b 

ix,y. 

IX         y 

Qx^ — X  y^—y 
ox-xy,^y 


Prob.  15.  Find  the  value  of  d  in  the  equation 

=  o.     (Ans.  d  =  7r/4.) 


sin  6^  sin  6^  o 
.101 
o     cos  6  cos  0 


Prob.  16.  Show  that  the  proportion  a\b\'.l\m  may  be  written 
—  o;  and  from  the  properties  of  this  determinant 


in  the  form 


Ifn 


prove  the  common  theorems  in  proportion. 
Prob.  17.  Show  that  the  determinant 


ab  c'  c' 
a^  be  a^ 
b'  b'  ca 


contains  the 


factor  {be  -\-  ea  ■\-  ab). 

Prob.  18.  Resolve  the  following  determinants  into  factors:* 

(i) 


(4) 


Art.  17.    Co-factors;  Minors. 
The  terms  oi  A  =  \ «/"'  |  which  contain  the  element  a^  may 


I  a  a^ 
lb  b' 

> 

(2) 

la  a" 
lb  b' 

a' 
b^ 

1 

(3) 

1  a^  a 
la^a 

;  .  .  .  ^, 

n-i 

U-I 

le  c" 

le  c"  e' 
id  d'd' 

lUnan     .   ,   .  an""-' 

I    I 

a  b 
a^b'' 
a*  b' 

I    I 

c  d 
c'  d' 
c'  d* 

> 

(5) 

I 

a 
a' 

I    I 

b    e 
b'e' 

> 

(6) 

III 

a  b   e 
a'  b'  c' 

be  obtained  by  expanding  the  determinant 


a,  a,    a. 


,a. 


(«) 


a'  a"  a'"        a  («> 


(1) 


For,  in  writing  out  this  expansion  each  term  is  formed  by 
taking  one,  and  only  one,  element  from  each  column  and  each 

*  These  determinants  belong  to  an  important  class  known  as   alternants. 
See  Hanus'  Elements  of  Determinants,  Boston,  1888,  pp.  187-201. 


22 


DETERMINANTS. 


row  of  the  array  (Art.  7).  If,  therefore,  in  selecting  the  ele- 
ments for  any  term,  any  other  element  than  a/  be  taken  from 
the  first  column,  the  one  taken  from  the  first  row  must  be  zero. 
Hence,  the  only  terms  which  do  not  vanish  are  those  which 
contain  the  element  a/. 

Moreover,  in  the  terms  of  the  expansion  of  (i)  which  do 
not  vanish,  a/  is  multiplied  by  {n  —  i)  elements  chosen  one 
from  each  column  and  each  row  of 


a''  a. 


.a„ 


an 


.  a, 


(n) 


(2) 


There  are  {n  —  i)!  such  terms,  any  one  of  which  may  be 
written  ±  a^al'aj"  .  . .  ^Z**^ ;  the  sign  being  determined  by  the 
class  of  the  permutation  of  the  n  subscripts  i,  i,j,  . .  .  I.  But 
since  this  is  of  the  same  class  as  the  permutation  of  the  {n  —  i) 
subscripts  /,/, . . .  /,  the  sign  of  any  term,  ±  a^aCaj"  . . .  ai^''\ 
of  the  expansion  of  (i)  is  the  same  as  the  sign  of  the  corre- 
sponding term,  al'a-" ..  .a^''\  of  the  expansion  of  (2).     H^nce, 


<2/  o 


o      ..  .0 


.a. 


-(«) 


{n) 


(«) 


(3) 


The  determinant  (2)  is  called  the  co-factor  or  complement 
of  the  element  «/  in  the  determinant  |<^/"^|.  It  is  obtained 
from  this  determinant  by  deleting  the  first  column  and  the 
first  row. 

The  co-factor  of  any  element  a^^^'^  maybe  found  in  the  same 
manner  upon  transposing  this  element  to  the  leading  position. 
But  by  this  transposition  the  sign  of  the  determinant  will  be 
changed  or  not  according  as  a^''^  occupies  a  negative  or  a  posi- 
tive position  (Art.  11).  Hence,  to  find  the  co-factor  of  any 
element  ^/*^  of  the  determinant  l^/**^],  delete  the  row  and  the 
column  to  which  the  element  belongs,  giving  the  resulting 
determinant  the  )  V^^  |  sign  when  (^  +  .)  is  {  -"  }  . 


DEVELOPMENT    IN    TERMS   OF    CO-FACTORS. 


25 


The  co-factor  thus  obtained  is  represented  by  the  symbol 

the  sign-factor  of  which,  (—  i)*+^  is  intrinsic,  i.e.,  included  in 
the  symbol  itself,  which  is  accordingly  written  as  positive. 
The  co-factors  of  the  various  elements  of  |  a^^a^^a^^  \  are  as 
follows : 


^n  = 

^38    ^33 

i 

^n-- 

^ai  ^a3 

^31    ^33 

y           A^^  = 

^,1  ^aa 
**^3i  **3a 

A..  =  - 

J 

^aa  = 

^n  ^13 

^3.    ^33 

»                 -^83  —   

^n  ^:a 
^31  ^3a 

A,,= 

f 

^3a^- 

;        -^33  ~ 

^ax  ^aa 

The  result  obtained  by  deleting  the  k\ki  row  and  the  jth 
column  of  J  ^  I  ^/"^  |  is  called  the  minor  of  the  determinant 
with  respect  to  the  element  dk't  and  is  written  A%.  This  minor 
is  the  same  as  the  co-factor  of  the  same  element  without  its 
sign-factor ;  thus : 

Similarly  A\i\k  is  the  result  obtained  by  deleting  the  ^th  and 
^th  rows  and  the  /th  and  ^th  columns  of  z/,  and  is  called 
a  second  minor  of  the  given  determinant.  Minors  of  still 
lower  orders  are  obtained  in  a  similar  manner,  and  expressed 
by  a  similar  notation.  The  >^th  minors  are  determinants  of 
the  order  {n  —  k). 


Art.  18.    Development  in  Terms  of  Co-factors. 

The  {n  —  i)\  terms  of  |  <3!/"^  |  which  contain  a^'^  are  repre- 
sented  in  the  aggregate  by  a^^'^Ak^'^  (Eq.  3,  Art.  17).  In  like 
manner  the  groups  of  terms  containing  the  successive  elements 
^k,  ^k' y '  •  •  ^A^"^  are  respectively 

Each  one  of  these  n  groups  includes  (n  —  i) !   terms   of  the 
determinant  |  ^/"^  |  ,  no  one  of  which  is  found  in    any  other 


24 


DETERMINANTS. 


group.  In  all  of  them,  then,  there  are  nX{^  —  i)l  or  n\  dif- 
ferent terms  of  the  determinant,  which  is  the  whole  number. 
Hence, 


(«) 


I  =  aM/  +  «."^/'  +  .  .  .  +  a.^'^W'. 


Similarly  (Art.  lo). 


(«) 


«/^M/^>  +  a.^'^A,^'^  +  . . .  +  a„^'^A„^^\ 


(I) 


(2) 


Any  determinant  may,  by  means  of  either  (i)  or  (2),  be  re- 
solved into  determinants  of  an  order  one  lower.  Since,  in 
these  formulas  A^y  .  .  .  Ak^"l  or  A^^'\  ,  .  .  A^'^  are  themselves 
determinants,  they  may  be  resolved  into  determinants  of  an 
order  still  one  lower  in  the  same  manner.  By  continuing  the 
process  any  determinant  may  ultimately  be  expressed  in  terms 
of  determinants  of  the  third  or  second  order,  which  may  be 
easily  expanded  by  methods  already  given  (Art.  9). 

For  example,  let  it  be  required  to  develop  the  determinant 
^  =  \  a^  d^  c,  d^  \  ,     Applying  formula  (i),  letting  k=  i,  gives 


A  =  a^ 


Upon  a  second  application  of  the  same  formula  this  becomes 


b,  c.  < 

-  b. 

a,  c,  d. 

+  ^. 

a^  b,  d. 

-< 

a^  b,  c^ 

b^  C,  d. 

^3  ^9  < 

a,  b,  d. 

^s  b,  c. 

b,  C,  d. 

a,  c,  d. 

ct,  b^  d. 

a,  b^  c. 

=  a,b. 

c,d. 

-  a,c. 

Kd, 

+  a,d. 

b.c. 

c,d. 

b,d. 

b,c. 

-a,K 

c,d. 

+  ^^, 

a,d. 

-b,d. 

a,  c. 

c,d. 

a,d. 

a,  c. 

+  a,c. 


—  a,d. 


I'.d, 

-Kc, 

a,d. 

+  c,  d. 

a,b. 

b.d. 

a  J, 

a,  b. 

b,c. 

+  b,d, 

a,c. 

-c,d. 

a>b. 

b.c. 

a,c. 

aj. 

The  complete  development  may  be  written  out  directly  from 
the  above.     It  is  given  in  Eq.  3,  Art.  7. 


THE    ZERO    FORMULAS. 


25 


Prob.  19.  Develop  the  following  determinants: 


(i) 


I  jc  17 
X  I  y  I 
ly  1  X 
y  I  X  I 


(2) 


a  X  y  a 
X  6  o  y 
y  Q  o  X 
a  y  X  a 


(3) 


o  q  r  s 
p  o  r  5 
p  q  o  s 
pqro 


Prob.  20.  Find  the  values  of  the  following  determinants: 


(i 


(4) 


1234 

2341 
3412 
4123 

0  I  I  I 

1  o  I  I 
I  I  O  I 
I  I  I  o 


(2) 


(s) 


0102 
1020 
0201 
2010 


(3) 


3 
6 

9  -3 

8   3 


5  3 

6  -I 

5 


3  3  3  3 
3222 
2  2  I  I 

I  I  I  o 

(Ans.  160;  9;  o; 


(6) 


0003 
1002 

O  I  O  I 

0010 

3;  -3;  -3) 


Prob.  21.  Obtain  the  determinants  in  Exs.  5  and  6  of  the  pre- 
ceding problem  from  that  in  Ex.  4. 


Prob.  22.  Evaluate 


oil. 
I  o  I  . 
I  I  o. 


,  of  the  «th  order. 

(Ans.  («  -  i)(-  i)«-' ) 


Prob.  23.  Show  that 


bed 
a  —d  c 
d       a    —b 


—  d   —c 


=  {a'-^-b'+c'+d'y. 


Art.  19.    The  Zero  Formulas. 

If  in  the  determinant  |  ^/"^  |  the  ^th  and  y^th  rows  be  sup- 
posed identical,  the  elements  ^/,  ^/',  .  .  .  a^'^'^  in  the  formula 
(i)  of  the  last  article  may  be  replaced  by  «/,  ^/',  .  .  .  U},^'''^  re- 
spectively. But  in  this  case  the  .value  of  the  determinant  is 
zero  (Art.  12).  Hence,  in  reference  to  the  determinant 
I  a^""^  I  ,  h  and  k  being  different  subscripts, 

aj!A^  +  a,''A^'  + . .  •  +  a.^^^A,^-^  =  o. 


26  DETERMINANTS. 

Similarly,  p  and  s  being  different  superscripts, 

^^w^^w  +  a^f^Ai^^  + .  .  .  +  ai^^A^^^^  =  o. 

Art.  20.    Cauchy's  Method  of  Development. 

It  is  frequently  desirable  to  expand  a  determinant  witli 
reference  to  the  elements  of  a  given  row  and  column. 

Let  the  determinant  be  ^  eee  |  a^^"^  \  ,  and  the  given  row 
and  column  the  ^th  and  /)th  respectively.  Then  is  A^^^^  the 
co-factor  of  «/^\  the  element  at  the  intersection  of  the  two 
given  lines.  The  co-factor  of  any  element  aj^^'^  of  A^^^  will  be 
designated  by  Bf,^'\  this  being  a  determinant  of  the  order 
(n  —  2).  The  required  expansion  may  now  be  obtained  by 
means  of  the  following  formula,  due  to  Cauchy : 

I  ^/«)  I  =  a.^'^Ai^^-  ^at^ai^^B,^^  (i) 

in  which  k  =  i,  2,. .  .^  —  i,  ^ -f-  I,  .  . .  «,  and  J  =  I,  2, .. ./  —  i, 

/+ i»  •••  ^>  successively. 

To  prove  this,  consider  that  B^'^  is  the  aggregate  of  all 
terms  of  the  expansion  of  A  which  contain  the  product 
ai^^ai'\  These  terms  are  included  in  a,i^^Aj}^\  Now,  every 
term  in  the  expansion  which  does  not  contain  a^^'^  must  contain 
some  other  element  a,i''^  from  the  ^th  row  and  also  some  other 
element  a^^'^  from  the  /th  column,  and  thus  contains  the  prod- 
uct alf^ai*^.  But  this  product  differs  from  ai^'^Uj,^'^  only  in  the 
order  of  the  superscripts ;  and  is,  therefore,  in  the  expansion  of 
A,  multiplied  by  an  aggregate  of  terms  differing  in  sign  only 
from  that  multiplying  ^^^^W^.  Hence,  —  a^^^W^Bk^''^  is  the 
coefficient  of  a^'^'a^^^  in  the  required  expansion. 

In  the  formula  a^^^Ai^^^  gives  (n  —  i) !  terms  of  J.      There 
are  also  («  —  i)"  such  aggregates  as  —  aJ^^^a^^'^Blf^^  each    con- 
taining («  —  2)  !  terms.     The  formula  therefore  gives 
{n  —  \y,-\-{n  —  \f  {n  —  2)\  =  n  \  terms,  which  is  the  complete 
expansion. 

When  the  expansion  is  required  with  reference  to  the  ele- 


CAUCHY  S  METHOD  OF  DEVELOPMENT. 


merits  of  the  first  column  and  the  first  row  the  formula,  written 
explicitly,  becomes 

{^  W|  =  a; a:-  a;a;'B;'  -  a:al"B;"  -  ...  -  a^a^-^B^-^ 
-  a^a^'B;'  -  a:a;"B^"  -  ...  -  a^a^^^B^-^ 


a:a^^^B^^\    (2) 


in  which  B^^'^  has  intrinsically  the  sign  (—  \f^\ 

Cauchy's  formula  is  particularly  useful  in  expanding  deter- 
minants which  have  been  bordered  ;  such  as 


-Q  = 


0 

u. 

7^2     «, 

«1 

«,, 

^,,^,3 

u. 

^.I 

^22   ^« 

u. 

^3. 

^32  ^33 

(3) 


Applying  formula  (2)  to  this  determinant  gives 

-  fi  =  -  «■' 


^23    ^,3 

+  «,«, 

a.. 

^« 

-     7/3^^, 

^21    ^35 

^32    ^33 

^3. 

^33 

^3,    ^3, 

^1,    ^,S 

-      ^.' 

^n 

^13 

+  M,U, 

^n   ^1, 

^8,    ^33 

^3, 

^33 

^31    ^35 

^1,    ^,3 

^u,u. 

^n 

^,3 

-K 

^n  ^n 

^«    ^.3 

^21 

^23 

^31    ^,Q 

Letting  ^^^  =  ^,^,  and, writing  ^4,1,  ^j,, ...  for  the  co-factors  of 
the  elements  of  |  ^1,^5,^3,  |  ,  the  above  becomes 

Prob.  24.  Develop    the    following    determinants    by   Cauchy's 
formula: 


(i)- 


a  h  g  u 
hb  fv 
gfcw 
u  vw o 


(2) 


0  yz  zx  xy 

;      (3) 

yz    0    I     I 

zx   I    0    1 

xy   1    1    0 

0  I       I      I 

1  o  xyzx 
I  .Tjv  oyz 
I  zx  yz  o 


28 

DETERMINANTS. 

(4) 

—  I 

—  X 

I 

I 

;     (5) 

1    1    I  X 

;    (6) 

O 

a        b 

I 

-  y 

—  I 

I 

X  y  z  o 

—  a 

sin^sin^ 

X 

o 

y 

z 

1  1  ly 

-b- 

-  cos^  cos^ 

I 

—  z 

I 

—  I 

I    I    I  z 

Art.  21.    Differentiation  of  Determinants. 
By  the  formula  (i)  of  Art.  i8 

A  ~  I  y,^^  I  =  Yk.yu,  +  Yk^yk^  +  . . .  F^^/^.  (i) 

Considering  the  elements  of  the  determinant  as  independent 
variables  and  differentiating  with  respect  to  yk^  gives 

Substituting  in  (i), 
Similarly 


d/i 


or 


dA 


Y   -^ 


=^-^^^^+-^^'^+ 


+J. 


dA 


dy, 


kn 


(2> 

(3> 


SA 


(4> 


Again  differentiating  (i),  this  time  with  respect  to  all  the  ele- 
ments of  the  k\.\\  row,  there  results 

S,A  =  Y,4y,.  +  Y,4y,,  + . . .  +  YkJy,^-  (5> 

In  the  total  differential  of  A  there  are  obviously  n  such  ex- 
pressions as  (5),  each  of  which  may  be  obtained  from  A  by 
replacing  the  elements  of  some  one  of  the  rows  by  their  differ- 
entials ;  thus : 


dA 


dy,, .  . .  dy^„ 
f.r  "•  y^n 

fm    '    '   '  ynn 


+ 


-y.n 

,dy. 


Jn, 


y. 


+  ...+ 


7n 

y.^ 


dy,,^  .,.dy,^ 


.(6> 


If  all  the  elements  are  functions  of  one  independent  variable  Xy, 

then,  representing  —^  by  j/^/, 
dx 


dA 
dx 


7i, 

y^x 


y.n 
y.n 

ynn 


+ 


7a/-.  •  yJ 


yr 


yr 


+...+ 


J.I 


y^n 
y.n 

-ynn 


.(7) 


RAISING    THE    ORDER. 


29 


Prob.  25.  Show  that  Cauchy's  formula  may  be  written 


dau'^'^ 


dak'^^da^- 


Art.  22.    Raising  the  Order. 

Since,  in  the  expansion  of  the  determinant  (i)  of  Art.  17 
the  elements  a^,  ,  . .  a^  do  not  appear,  these  may  be  replaced 
by  any  quantities  whatever,  as  Q,  . ,  ,  Ty  without  changing  the 
value  of  the  determinant  ;  thus: 


^,     o      o    . . .    o 

a:  a:'  a:"  . . .  ^,(«) 


^/  o     o    . . .    o 


Similarly, 


^/   0     0 
a  J  aJ'    0 

. ..    0 

...     0 

=«' 

.  .  «,"" 

al  0    0  ...    0 
Qa^'  0  ...    0 

M'J         l-^J                    \J 

a:". 

.  .  «i"' 

a:  a:'  a:" 

. . .  «„'"' 

TNaJ"...ai''^ 

in  which  Q,  R,  , . .  T  and  Z, .  . .  iVare  any  quantities  whatever. 

Finally, 

o    =«'... ^«ir^W"^=  <  o   ...  o      o 
o  QaJ'...  o      o 


a^       o      .  . .  o 
a^     a^       .  . .  o 


^'«-x^Vx...^jr^o 


^«        ttn 


.  .   ^  <«-^>    ^  <«> 


s  M . .  .a^^:^r^  o 


that  is,  if  all  the  elements  on  one  side  of  the  principal  diagonal 
are  zeros'the  determinant  is  equal  to  its  principal  term,  and 
the  elements  on  the  other  side  of  this  diagonal  may  be  replaced 
by  any  quantities  whatever. 
By  what  precedes, 

/  .  . .  ^ .<«> 


«„'...  a 


(«) 


10... 

0 

Qa:... 

«,<"> 

Taj... 

«»"" 

30 


DETERMINANTS. 


Hence,  a  determinant  of  the  nth.  order  may  be  expressed 
as  a  determinant  of  the  order  (« -|-  i)  by  bordering  it  above 
by  a  row  (to  the  left  by  a  column)  of  zeros,  to  the  left  by  a 
column  (above  by  a  row)  of  elements  chosen  arbitrarily,  and 
writing  i  at  the  intersection  of  the  lines  thus  added.  By  con- 
tinuing this  process  any  determinant  may  be  expressed  as  a 
determinant  of  any  higher  order, 

Prob.  26.  If  all  the  elements  on  one  side  of  the  secondary  diag- 
onal are  zeros,  what  is  the  value  of  the  determinant  ? 


Prob.  27.  Develop  the  determinant 


a 

h 

g 

u 

0 

h 

b 

f 

V 

0 

S 

f 

c 

w 

0 

u 

V 

w 

0 

t 

0 

0 

0 

/ 

s 

Prob.  28.  A  determinant  in  which  ^^^^^  =  —  as^^^  and 


ak 


[k) 


O  IS 


said  to  be  skew-symmetric.     Prove  that  every  skew-symmetric  deter- 
minant of  odd  order  is  equal  to  zero. 


Art  23.    Lowering  the  Order. 

The  following  method  of  reducing  and  evaluating  a  deter- 
minant is  often  useful,  particularly  when  the  elements  are  numer- 
ical.    Let  the  determinant  be 


J  = 


ai'a/' 


ai 


in) 


J  = 


I 


Then 


(«iO 


/\n-l 


^2      0L\  ^2       Ci\  (I2 

«3'  ai'as"  ai'aa"' 


(«i') 


An-2 


(1\  OL2     — ^2  ^1       ^1  ^2       — ^2  t*! 


a^ar 


(Art.  13.) 


(Arts.  15, 17.) 


In  this  last  expression  the  determinant  is  obviously  of  the 
order  (w  — i).  The  process  may  be  formulated  thus:  Replace 
each  element  a)^^^  of  the  minor  of  the  leading  element  by  the 


SOLUTION   OF  LINEAR  EQUATIONS. 


31 


determinant 


and  the  resulting   determinant  divided 


by  (ai')'»~2  will  be  equal  to  the  given  determinant.  The  elements 
being  numerical  the  process  may  be  repeated  with  ease  until 
the  order  becomes  unity.    The  example  given  below  will  illusffat 


1234 

= 

I  2 

I  3 

I  4 

8765 

87 

8  6 

85 

1827 

3645 

I  2 
I  8 

I  3 
I  2 

I  4 

I  7 

I  2 

I  3 

I  4 

36 

3  4 

3  5 

-9    -18    -2) 

'    = 

-9  -18 

- 

-9 

-27 

6-1         . 

' 

6   -   I 

6 

3 

0  -  5   -   / 

-9  -18 

- 

-9 

-27 

1 

0  -  5 

0 

-  7 

117  135 

=  1296. 

45    63 

Art.  24.    Solution  of  Linear  Equations. 

Of  the  many  analytical  processes  giving  rise  to  determinants 
the  simplest  and  most  common  is  the  solution  of  systems  of 
simultaneous  linear  equations.     Thus,  solving  the  equations 

a.V  +  a:'x"  =  /<•„  ) 
a:x'  +  a,"x"  =  K,,  \ 

by  the  methods  of  ordinary  algebra  gives : 


X'  = 


In  the  notation  of  determinants  these  are  written : 


K,a,''  -  K,a/' 

x" 

«i  «t  —  c'i  ^t 

« 

X    = 


/ 


a,  a- 


/ 


It  will  be  noted  that  the  two  fractions  expressing  the  values 
of  x'  and  x'^  have  a  common  denominator,  this  being  the  de- 
terminant whose  elements  are  the  coefficients  of  the  unknowns 
arranged  in  the  same  order  as  in  the  given  equations-     The 


32 


DETERMINANTS. 


numerator  of  the  fraction  giving  the  value  of  x'  is  formed  from 
this  denominator  by  replacing  each  coefificient  of  x'  by  the 
corresponding  absolute  term.     Similarly  for  x" . 

The  difficulty  of  solving  systems  of  linear  equations  by  the 
ordinary  processes  of  elimination  increases  rapidly  as  the  num- 
ber of  equations  is  increased.  The  law  of  formation  of  the 
roots  explained  above  is.  however,  capable  of  generalization, 
being  equally  applicable  to  all  complete  linear  systems,  as  will 
now  be  shown. 

Let  such  a  system  be  written 


a;x'  +  a^'x"  +  . . 


4-  <3:/«';ir(«'  =  K^ , 


a:x'  +  a^'x"  +  .  .  .  +  ar'x^'''  =  k„. 


(I) 


Now  form  the  determinant  of    the  coefficients  of    these 
equations ;  thus : 


n 


,a, 


(«) 


and  let  A^^  be  the  co-factor  of  ai''^  in  this  determinant.  The 
function 

is  equal  to  D  when  p  —  s  (Art.  i8) ;  to  zero  when  /  and  s  are 
different  superscripts  (Art.  19).  Then,  multiplying  the  given 
equations  by  y4/*\  A^""^^  .  .  .  Aif^  respectively,  the  sum  of  the 
resulting  equations  is  a  linear  equation  in  which  the  coefficient 
of  x^^^  is  equal  to  D,  while  those  of  all  the  other  unknowns 
vanish.     The  sum  is,  therefore. 


Dx^''  =  /c,^/^>  +  K,Ai^^  +  .  .  .  +  K^A, 


(2) 


But  the  second  member  of  this  equation  is  what  D  becomes 
upon  replacing  the  coefficients  «/*\  a^^^, .  . .  a^f^  of  the  unknown 
;ir<*'  by  the  absolute  terms  /c, ,  /c, ,  .  .  .  /c«  in  order.     Hence, 


-W  — 


a/. 

CONSISTENCE    OF    LINE 

/ 

SYSTEMS, 
a.'a,"  .  . 

33 
(3) 

a.'.. 

.aJi^-^^K^ai^^^K,.a^^^ 

«>.".. 

•  «„'"' 

This  result  may  be  stated  as  follows  : 

{a)  The  common  denominator  of  the  fractions  expressing 
the  values  of  the  unknowns  in  a  system  of  n  linear  equations 
involving  ;/  unknown  quantities  is  the  determinant  of  the 
coefficients,  these  being  written  in  the  same  order  as  in  the 
given  equations,  {b)  The  numerator  of  the  fraction  giving  the 
value  of  any  one  of  the  unknowns  is  a  determinant,  which  may 
be  formed  from  the  determinant  of  the  coefficients  by  sub- 
stituting for  the  column  made  up  of  the  coefficients  of  the 
unknown  in  question  a  column  whose  elements  are  the  absolute 
terms  of  the  equations  taken  in  the  same  order  as  the  coeffi- 
cients which  they  displace. 

Prob.  29.  Solve  the  following  systems  of  equations  : 


(l) 

3^  +  5^=21.     6^  +  27=15; 

(2) 

f  +  f  =  s,   f+.  =  .= 

(3) 

3^  4-  7  +  22  =  50,     ^  +  2j;  -  32:  =  15, 

(4) 

I    1    I        ^      ^     I    ^                ^     I     ^ 

(5) 

w    ,    X    .     y  .    z          ^           w    .   X 

w    ,   X    .    y    .     z 

7  +  9  +  "  +  ^  =  ''^^' 


-+^  =  2144, 

-  +  -  +  ^  +  -  =  1472. 
9     II     13     15 


Prob.  30.  Show  that  the  three  right  lines 

y  =  x-\-  I,    y  =  —  2x  +  i6y    y  =  sx  —  gy 
intersect  in  a  common  point. 


Art,  25.    Consistence  of  Linear  Systems. 

When  the  number  of  given  equations  is  greater  than  the 
number  of  unknowns  their  consistency  with  one  another  must 


34 


DETERMINANTS. 


obviously  depend  upon  some  relation  among  the  coefficients. 
This  relation  will  now  be  investigated  for  the  case  of  {n  -\-  i) 
linear  equations  involving  n  unknowns.     Let  the  equations  be 


^,V 


+  ...  +  a. 


W^yW       _ 


«■..  ^ 


««v 


+  ...  +  a. 


(«)  -yin) 


^n\ 


^«+x  V  +  .  .  .  +  <U(«)  =  /C,^,.        J 

If  the  above  equations  are  consistent  the  values  of  the  unknowns 
obtained  by  solving  any  n  of  them  must  satisfy  the  remaining 
equation.  Solving  the  first  n  equations  by  the  method  of  the 
preceding  article,  substituting  the  values  of  x\  x"y . . .  x^""^  thus 
obtained  in  the  last  equation,  and  clearing  of  fractions,  the 
result  reduces  to  (Art.  i8) 


. . .  ^/«) 


/c. 


«+i 


(«) 


(«) 


K 


M+I 


=  0, 


which  is  the  condition  to  be  fulfilled  by  the  coefficients  in  order 
that  the  given  equations  may  be  consistent.     ^ 

Hence  the  condition  of  consistency  for  a  set  of  linear  equa- 
tions involving  a  number  of  unknowns  one  less  than  the  number 
of  equations  is  that  the  determinant  of  the  coefficients  and 
absolute  terms,  written  in  the  same  order  as  in  the  given  equa- 
tions, shall  be  zero.  This  determinant  is  called  the  resultant* 
or  eliminant  of  the  equations.     Thus  the  equations 

X ~\-y  —  z  =  o,  X— y-\'Z=2,  —  X -\-y -{-2  =  4,  x  +7 -{-s  =  6 

are  consistent,  for  the  reason  that 


I  I  —I 

I  —I  I 

—  I  I  I 

I  I  I 


=  0. 


*  This  term  was  introduced  by  Lapla"ce  in  1772. 
due  to  Sylvester. 


The  term  eliminant  is 


HOMOGENEOUS    LINEAR    SYSTEMS. 


35 


Art.  26.    The  Matrix. 
Assume  ,r  linear  equations  involving  n  unknowns,  r  being 
greater  than  n.  as  follows : 


a:x'-\-,.,-\-a 


(«)^(«) 


'^M» 


a^x'  +  . . .  +  ^/");ir(«)  =  Kr. 
The  consistency  of  these  equations  requires  that  every  deter- 
minant of  the  order  {ii  +  i),  formed  by  selecting  (n  +  i)  rows 
from  the  array  whose  elements  are  the  coefificients  and  abso- 
lute terms  written  in  order,  shall  be  zero. 

If  the  elements  of  the  array  fulfill  this  condition  the  fact  is 
expressed  thus : 

al   ...^„'   ...V     =0; 


K,      .  .  .  K^ 


the  change  of  rows  into  columns  being  purely  arbitrary.     The 
above  expression  is  called  a  rectangular  array,  or  a  matrix. 


Art.  27.    Homogeneous  Linear  Systems. 

Let  the  equations  of  the  given  system  be  both  linear  and 
homogeneous  ;  thus  : 

a^  x'  -\-  ,  ,  ,-\-  ^/«);t;('')  =  o,  " 


(I) 


a:  x'-\-,\.+  ^J«>;ir(«>  =  O. 

Representing  the  determinant  of  the  coefficients  by  E,  the 
general  solution,  as  given  by  the  formula  (3)  of  Art.  24,  is 

;ir(^)  =  O/E, 

That  is,  all  the  unknowns  are  equal  to  zero,  and  the  equa- 
tions have  no  other  solution  than  this  unless 

E  =  o. 


36 


DETERMINANTS. 


But  in  this  case  the  value  of  each  unknown  is  obtainable 
only  in  the  indeterminate  form  o/o.  The  ratios  of  the  un- 
knowns may  be  readily  obtained,  however.  For,  dividing  each 
equation  through  by  any  one  of  these,  as  x^'\  the  system  (i) 

becomes 

(*-i) 

I-   (2) 


a'- I-      _i-^(*-i)f _L^(s+i)::r l      4_/7(«)_  — _/7(^) 


•  ^>+ ■■■+-• 


is -I) 


X^^ 


+a 


is+iY 


'(S+l) 


is) 


+  ...+a 


(n) 


;r<'') 


,(s) 


J 


Now  the  condition  E  =  o  establishes  the  consistency  of  the 
n  equations  (2)  involving  the  {n  —  i)  unknown  ratios  (Art.  25), 


-(*)' 


.(s+i) 


An) 
Is)' 


Hence,  if  ^  =  o'the  given  equations  (l)  are  consistent ;  that  is, 
the  values  of  the  above  {n  —  i)  ratios  obtained  by  solving  any 
(n  —  i)  of  them  will  satisfy  the  remaining  equation.  Any  n 
-quantities  having  among  themselves  the  ratios  thus  determined 
will  satisfy  the  given  equations.  Thus,  if  x^\  x^\  .  .  .  x^'^^  are 
n  such  quantities,  so  also  are  \x^,  A  x^' ,  .  .  .  Ajf/'*^  A  being  any 
factor  whatever. 

The  determinant  E  of  the  coefificients  of  the  given  homo- 
geneous linear  equations  is  called  the  resultant  or  eliminant  of 
the  system. 

When  the  number  of  equations  is  greater  than  the  number 
of  unknowns  the  conditions  of  consistency  are  expressible  in 
the  form  of  a  rectangular  array,  as  in  Art.  26. 

As  an  example,  consider  the  five  equations 

2x-  iy^z-=^o,    4x  -j/  —  ^  =  o,     -  7-^  +  37  +  ^  =  o, 
x-]-}>  —  js  =  o,      t^x  —  ^y -\- z  =  o. 
Dividing  each  of  the  first  two  equations  by  z  and  solving 


X  y 

Tor  the  two  unknowns  -  and  -  gives 
z  z 


—  I 
I 


2  —  I 
14        I 


2-3 

4—  I 


^3^ 

5 


CO-FACTORS    IN    A    ZERO    DETERMINANT. 


37 


or  x:j/:2::2:siS'i 

and  any  three  quantities  having  these  ratios  will  satisfy  the 

two  equations,  as  lo,  15,  and  25.     That  the  third  equation  is 

consistent  with  the  first  two  is  shown  by  the  vanishing  of  the 

determinant 

2  —  3       I 

4-  I  -  I 

-73       I 

If  all  the  equations  are  consistent  the  determinant  of  the 
coefficients  of  any  three  of  them  must  vanish ;  that  is, 


2 

4-7 

I 

5 

-3 

-  I       3 

I 

-5 

I 

—  I       I  — 

I 

I 

Art.  28.     Co-factors  in  a  Zero  Determinant. 

If,  in  the  preceding  article,  £  =  o,  it  follows  from  Arts.  i8 
and  19  that 

a/ A,'  +  a,"  A,"  +  ...+  <?,<"'^.<"'  =  o, 


WA,'  +  a,"  A,"  +  ...+  a,('W^  =  o  =  E, 

a„'A,'  +  a„"A,"  +  ...  +  a.^'W"'  =  O. 
These  equations  obviously  give  for  the  ratios 


A/ 


A,^^^' 


At'  ' 


A^-^ 


At'  '     At^  '   "    At' 
values  which  are  identical  with  those  obtained  for  the  ratios 

from  the  equations  (i)  of  Art.  27.  It  follows  that  x',  x" ,  . .  .  x^''"> 
are  proportional  to  ^/,  A^' ^  .  .  .  Af'^,  whatever  the  value  of  k. 
Thus,  giving  to  k  the  successive  values  i,  2,  .  .  .  n,  there  result 

x'  \x"  \.,,  :x^"^::A/:A,'':.  .  .  i^/'*) 

i'.AJ'.A:':...  :^,<«\ 


: :  AJ  :  A, 


AJ-l 


38 


DETERMINANTS. 


Hence,  when  a  determinant  is  equal  to  zero,  the  co-factors 
of  the  elements  of  any  line  are  proportional  to  the  co-factors  of 
the  corresponding  elements  of  any  parallel  line. 


Art.  29.    Sylvester's  Method  of  Elimination.* 

Let  it  be  required  to  eliminate  the  unknown  from  the  two 

equations 

a^x^  +  ^i^^  -\-  a^x  -{-  a^  =  o, 

d,x'  +  b^x  -\-b,-=Q. 

This  will  be  done  by  what  is  called  the  dialytic  method,  the 
invention  of  which  is  due  to  Sylvester.     Multiplying  the  first 
of  the  given  equations  by  x^  and  the  second  by  x  and  x"^  suc- 
cessively, the  result  is  a  system  of  five  equations,  viz.: 
a^x^  +  a^x""  -)-  a^x  -}-  ^„  =  o, 

b^x''  +  b^x'  +  b,x  =  o, 

b,x\+ b,x' +  b,x'  =0. 

The   eliminant   of    these   five   equations,   involving  the   four 
unknowns  x,  x\  x\  and  x*  is  (Art.  25) 


£  = 


0 

a. 

^. 

a^ 

^0 

a. 

^h 

«, 

^0 

0 

0 

0 

K 

K 

^ 

0 

K 

b^ 

K 

0 

K 

b. 

K 

0 

0 

=  0. 


If  the  given  equations  be  not  consistent  this  determinant  will 
not  vanish. 

The  above  method  is  a  general  one.     Thus,  let  the  two 
given  equations  be 

a^x^^  +  . . .  +  /3!,;ir  +  ^„  =  o, 

^„^"-f +  ^,^  +  ^0=0. 

Multiplying  the  first  equation  (;2  —  i^  times  in  succession 
by  x^  and  the   second  {m  —  i)  times,  {m -\- ti)   equations  are 

*  Philosophical  Magazine,  1840,  and  Crelle's  Journal,  Vol.  XXI. 


SYLVESTER  S    METHOD    OF    ELIMINATION. 


3^ 


obtained  which  involve  as  unknowns  the  first  {m  -\-  n  —  i) 
powers  of  x.  The  eliminant  of  these  equations  is  a  determinant 
of  the  order  {m  -\-  n),  which  is  of  the  nth  degree  in  terms  of 
the  coef^cients  of  the  equation  of  the  mth  degree,  and  vice 
versa.     The  law  of  formation  of  the  eHminant  is  obvious. 

The  same  method  may  be  used  in  eliminating  one  or  both 
the  variables  from  a  pair  of  homogeneous  equations. 

As  an  example,  let  it  be  required  to  eliminate  the  variables 
from  the  equations 

2x^  —  ^x^y  —  9jj/^  =  o     and     '^x'^  —  yxy  —  6y  =  o. 

X 

Dividing  the  first  byjJ/^  and  multiplying  by  — ;  the  second 

X 

byy,  and  multiplying  by—  twice  in  succession,  there  result, 


X    X      X  X 

in  all,  five  equations  involving  — ,  — ,  -^,  and  -j- 
these  four  ratios  gives 


Eliminating 


E~ 


5  o- 9 
0  —  9  o 
3-7-6 
7  —  6  o 
600 


the  vanishing  of  which  shows  that  the  two  given  equations  are 
consistent. 

Prob.  31.  Test  the  consistency  of  each  of  the  following  systems 
of  equations: 

(i)  ^+^+22=9,  x-\-y  —  z=o,  2x—y  +  z=^,  x—3y+2z=i; 
(2)  X  —y  —  2Z  =0,       X  —  2y  •\-  z  =  o, 


2X 


sy 


o; 


(3)  2^y  —  ^y  =  o,       8^  jV  +  Sxy'  —  5/  =  o. 

Prob.  32.  Find  the  ratios  of  the  unknowns  in  the  equations 

2X  -{-y  —  2z  =  o,     4W  —  y  ^  4Z  =  o,     2w  -{-  x  —  ^v  -\-  z  =  o. 

Prob.  33.  In  the  equations 

ak'x'  +  .  .  .  +«a('')jc(«)  +  ^^(«  +^)^(«+i)  r=  o,     [^  =  I,  2,  .  .  .  «] 
prove    that   ^' :  .  .  .  :  x^""^  :  ^(«+^> ::  J/' :  .  .  .  :  J/^«> :  J/(«+i) ,   where 


4-0 


DETERMINANTS. 


{— i)'"'-^^'^  is  the  determinant  obtained  by  deleting  the  zth  column 
irom  the  rectangular  array 


M  = 


aWa^(n+^) 


an  .  .  ,  Un       ^H 


lx-\-yy  -\-  i^z  _vx  -\-  my  -\-\z  _  ^x-\-\y  +  nz 
P  ~  ■ 


X 


y      _ 


y  /// 

I  ^p 

Iv  p 

m\  q 

V  \  q 

V  m  q 

\  n  r 

}J.  n  r 

)xX  r 

Prob.  34.  From 
deduce 


Prob.  35.  Show  that  the  three  straight  lines  dt':r  -|-  b'y  -f-  /  =  o, 
ta^'x  +  b"y  +  /'  =  o,    and  a"'x  +  b'"y  +  c'"  =  o,  are  concurrent 
when  I  ^rV"  |  =  o. 

Prob.  36.  Prove  that  the  medians  of  a  triangle  are  concurrent. 

Prob.  37.   Show  that  the  points  (x^,y^),  (x, ,}>,),  and   (x^,y^)   are 
collinear  when    x^  y^  i     =  o. 


•^0 

y. 

I 

X, 

y^ 

I 

^, 

y. 

I 

Prob.    38.    Write    the    conditions    that    all    the   points    (^,,jKi), 
■{x^,y^),  .  .  .  (xn,yn)  shall  be  collinear  in  the  form  of  a  matrix. 

Prob.  39.  Obtain  the  equation  of  a  right  line   through  (x^^yj 
.and  (x^,y^)  in  the  form  of  a  determinant. 

Prob.  40.  Show  that  the  equation    x   y    z    j 

X,  y,  z^  I 
x^  y,  z^  I 

^3  y,  ^3  I 

'represents  a  plane  through  {x^ ,  y^ ,  z^),  (x^ ,  y^ ,  z^),  and  (^g ,  y, ,  z^). 

Art.  30.    The  Multiplication  Theorem. 
Let  the  two  homogeneous  linear  equations 


.•be  subjected  to  linear  transformation  by  substituting 


(I) 


(2) 


THE    MULTTPLICATION    THEOREM. 


41 


(3) 
(4) 


The  result  of  such  transformation  is 

(a  J,,  +  aj.^u,  +  {aj^,  +  aj^^u. 
The  vanishing  of  the  determinant 

is  the  condition  that  the  equations  (3)  may  be  consistent ;  that 
is,  the  condition  that  they  may  have  solutions  other  than 
u^  =z  o  =z  u^  (Art.  27).  Now  the  equations  (3)  may  be  consist- 
ent because  of  the  consistency  of  the  equations  (i),  in  which 
case  the  determinant 

«..  a,,         ,  (5) 

vanishes.  Or,  this  condition  failing,  and  the  equations  (i) 
thus  having  no  solution  other  than  x^=z  o  ^=  x^y  the  equations 
(3)  will  still  be  consistent  if  the  equations  (2)  are  so ;  that  is, 
if  the  determinant 


^11   ^X1 


(6) 


vanishes.  The  vanishing  of  either  of  the  determinants  (5)  or 
X6),  therefore,  causes  the  determinant  (4)  to  vanish.  It  follows 
that  (5)  and  (6)  are  factors  of  (4) ;  and  since  their  product  and 
the  determinant  (4)  are  of  the  same  degree  with  respect  to 
the  coefficients  <?„,...,  /^^ ,  ... ,  they  are  the  only  factors. 
Hence, 


(7) 


The  above  method  is  equally  applicable  to  the  formation 
•of  the  product  of  any  two  determinants  of  the  same  order. 
Hence  results  the  following  general  formula: 

I  a\\  a-21  .  .  .  a^n  |   '   |  ^n  <^22  .  .  .  bnn  I    = 


a\\bw\- '.<  ■\-a\yJ)in     aiid'ii+ . . .  +aind2n  •  .  .  .  aiidni+ . . .  +ainbnn 

aiidii+  .  .  .  +a2nbin      atidii  +  .  .  .  -\- a2n^2n  •    •    •    .  ^2i<^ni+  ..  .   +^2n<5nn 
Clnibii-\-  ...  -{-annbin      ««i<^2i  +  .  .  .  +  ^nn<^2n  .    .    .    .  am^ni  +  .  .  .  +  annKn 


(8) 


42 


DETERMINANTS. 


The  process  indicated  by  this  formula  may  be  described  as 
follows :  * 

To  form  the  determinant  |/,^„|,  which  is  the  product  of 
two  determinants  \a^^„\  and  |<^,,„|,  first  connect  by  plus  signs 
the  elements  in  the  rows  of  both  |<^j„|  and  |<^j_„| .  Then  place 
the  first  row^  of  \a^,„\  upon  each  row  of  |^,„|  in  turn  and  let 
each  two  elements  as  they  touch  become  products.  This  is 
the  first  row  of  |/,,«|.  Perform  the  same  operation  upon  \d^^„\ 
with  the  second  row  of  \a^^„\  to  obtain  the  second  row  of  |/j_„| ; 
and  again  with  the  third  row  of  |^,„|to  obtain  the  third  row 
of|/,,«|;  etc. 

Any  element  of  this  product  is 

As  =  ^hK  +  ^k,K  +  •    •   •    +  ^kn^sn-  (9> 

When  the  two  determinants  to  be  multiplied  together  are 
of  different  orders  the  one  of  lower  order  should  be  expressed 
as. a  determinant  of  the  same  order  as  the  other  (Art.  22),  after 
which  the  above  rule  is  applicabje. 

The  product  of  two  determinants  may  be  formed  by 
columns,  instead  of  by  rows  as  above.  In  this  case  the  result 
is  obtained  in  a  different  form.  Thus  the  product  of  the  de- 
terminants (5)  and  (6)  by  columns  is 

Prob.  41.  Form  the  following  products  : 


(3) 


(i)  la  A  g 
h  b  f 
g  f  c 
^1,  ^12  ^.3 

^21    ^22    ^23 
^»1    ^32    '^SS 


\g  c 


(2) 


b  f 
f  c 


^  g 

g   c 


a  h 
h  b 


^)2    ^13 


A 

4  4       4 

21  22  23 

-^31  -^32    -^33 


;      (4) 

a,  b,  c. 

. 

oil 

a,  b,  c^ 

I    0    I 

«3  K  ^3 

I    I    0 

Prob.  42.  Generalize  the  last  example  (see  Prob.  22,  Art.  18). 
Prob.  43.  By  forming  the  product 


a-bV- 


I  ■\-  m  V —  I 


l-\-mV-  I 

j-  k  V~^^ 


*  Carr's  Synopsis  of  Pure  Mathematics,  London,  18S6,  Article  570. 


PRODUCT    OF    TWO    ARRAYS. 


43 


ishow   that   the   product   of   two  numbers,  each  the  sum  of  four 
squares,  is  itself  the  sum  of  four  squares. 

Art.  31.    Product  of  Two  Arrays. 

The  process  explained  in  the  preceding  article  may  be  ap- 
plied to  form  what  is  conventionally  termed  the  product  of 
two  rectangular  arrays.  It  will  appear,  however,  that  multi- 
plying two  such  arrays  together  by  columns  leads  to  a  result 
radically  different  from  that  obtained  when  the  product  is 
formed  by  rows. 

Let  the  two  rectangular  arrays  be 


^n^iQ^is 


and 


b^.b^^b^^ 


The  product  of  these  by  columns  is  / 

^11^12  +  ^^/sQ     ^i^b,^  +  a^^b^^     a^J}^^  +  «„^„ 

The  determinant  ^  is  plainly  equal  to  zero,  being  the  prod- 
uct of  two  determinants  formed  by  adding  a  row  of  zeros  to  one 
of  the  given  rectangular  arrays  and  a  row  of  elements  chosen 
arbitrarily  to  the  other. 

In  general,  the  product  by  columns  of  two  rectangular 
arrays  having  m  rows  and  n  columns,  m  being  less  than  Uy  is  a 
determinant  of  the  n^^  order  whose  value  is  zero. 

Multiplying  together  the  above  rectangular  arrays  by  rows, 
the  result  is 


A'^ 

^n 

6,,  +  a 

,/.. 

+   ^13^,3 

a.Ar 

+  a 

12^22   +   ' 

^13 

5„ 

^2/11   +  ^22<^I2   +   ^23<^13                   <3:3,/53,    +    ^„^„   +   ^„^„ 

a,,a,. 

. 

6Jn 

+ 

^,1^13 

. 

bj.. 

-U 

^n^i2 

. 

ij. 

^22^23 

6.A, 

^21^2  3 

b.A. 

^21^,2 

b 

,A,  1 

In  the  same  manner  it  may  be  shown  that  the  product  by 
rows  of  two  rectangular  arrays  having  m  rows  and  n  columns, 
m  being  less  than  «,  is  a  determinant  of  the  m^^  order,  which 
may  be  expressed  as  the  sum  of  the  n  \/m !  {n  —  w)  [determinants 


44 


DETERMINANTS. 


formed  from  one  of  the  arrays  by  deleting  {n  —  m)  columns,, 
each  multiplied  by  the  determinant  formed  by  deleting  the 
same  columns  from  the  other  array. 

Art.  32.     Reciprocal  Determinants. 

The  determinant  formed  by  replacing  each  element  of  a 
given  determinant  by  its  co-factor  is  called  the  reciprocal  o£ 
the  given  determinant.*     Thus,  the  reciprocal  of 


^n^„ 


IS 


A 


-^ai-^23  .  .   .  A^n 


The  product  of  these  two  determinants  is 


S.^= 


aiiAn-\-  .  .  .  -\-ainAin     flii^2i+  .  .  .+«in^3n 


CliiAni'\-'  .  '-{-a-inAnn 


aniAii-j-.  .  .  .-{-annAm     «nMai+-  •  .+«n»'43n.  •  .  .  aniAni+.  •  -+annAnn 

Each  element  on  the  principal  diagonal  of  this  product  is 
equal  to  6  (Art.  i8),  while  all  the  other  elements  vanish  (Art, 
19).     Hence, 


d,A  = 


6  o.  . 
o   d  .. 


Q(n) 

o 


=  (J%     or    ^ 


o„  o  .  .  .  d 

That  is,  the  reciprocal  of  a  determinant  of  the  n^^  order  is 
equal  to  its  {n  —  i)*^  power. 

Art.  33.    Linear  Transformations. 
Let  it  be  required  to  transform  the  system 

ahiXi+ah2X2  +  .  .  .+ahnOCn  =  o     [A=I,  2,  ...w]  (l) 

into  a  new  system  with  the  variables  Wi,  W2j  •  •  •  ^n  by  means  of 
the  linear  substitutions 


QUANTICS:    INVARIANTS   AND   COVARIANTS.  45 

Making  the  required  transformation,  as  has  been  done  for  the 
case  of  two  variables  in  Art.  30,  the  resulting  system  is 

PjlUl  +pj2U2  +  .  .  .  +pjnUn  =  0,      [j  =  I,  2,  ...  w]  (3). 

in  which  pks  =  c^hKi  +  (^k2bs2  + .  .  .  +  aknbsn-     (Eq.  9,  Art.  30.) 

The  determinants  of  the  systems  (i),  (2),  and  (3)  are  thus  con- 
nected by  the  relation 

l/^l.nhl^l.nl.l^l.nl.  (4) 

The  determinant  |  ^i.n  |>  whose  elements  are  the  coefficients 
in  the  equations  (2),  is  called  the  modulus  of  transformation; 
and  the  relation  expressed  by  equation  (4)  may  be  stated  as 
follows : 

If  a  system  of  n  homogeneous  linear  equations  in  n  variables 
be  subjected  to  linear  transformation,  the  eliminant  of  the  trans- 
formed equations  will  be  the  eliminant  of  the  given  equations- 
multiplied  by  the  modulus  of  transformation.  A  transformation 
whose  modulus  is  unity  is  said  to  be  unimodular. 

Prob.  44.  Show  that  the  following  transformations  are  unimodular: 

(i)  x=x'+y^+2z',    y=x'-\-y'-\-z'^     z=y'-\-z'\ 

(2)  ^=:x:' cosa— y  sina,    >'=x' sinaH-^'' cosa. 

Art.  34.    Qu antics;   Invariants  and  Covariants. 

A  homogeneous  function  of  any  number  of  variables  is  called 
a  quantic. 

A  quantic  is  binary,  ternary, .  . .  w-ary  according  as  it  con- 
tains two,  three, ,  ,  .n  variables;  and  is  specifically  known  as  a 
quadric,  cubic, . . .  w-ic  according  as  it  is  of  the  second,  third, 
. . .  wth  degree.     Thus,  the  function 

q=  aiiXi^  +  ^22^2^  +  ^33^3^  +  2a23^2^3  +  2azxXzXY  +  2a\  2^1  rv2 

is  a  ternary  quadric. 

A  covariant  is  a  quantic  derived  from  another  quantic  in  such 
manner  that  when  both  are  transformed  by  the  same  linear  sub- 
stitutions the  resulting  quantics  are  still  connected  by  the  same 
process  of  derivation. 


46 


DETERMINANTS. 


An  invariant  is  a  function  of  the  coefficients  of  a  quantic 
which  is  not  effected  by  linear  transformation  of  the  quantic, 
except  that  it  is  multiplied  by  a  power  of  the  modulus. 

It  is  obvious  that  every  invariant  of  a  covariant  is  an  invariant 
of  the  original  quantic. 

Art.  35.    The  Discriminant. 

The  discriminant  of  a  quantic  is  the  eliminant  of  its  first 
derivatives. 

Thus,  the  binary  quadric 

(f)=  aiiXi^  +  ^22^2^  +  2ai2^i:x:2=o 
gives 

1  30  ,  130 

—x —  =  aii^i  +  ai2^2=o,    —7^ —  =  ^1 2X1+^22^2=0. 

2  OXi  2  0x2 

Hence,  writing  ai2  =  (i2i,  the  discriminant  is 

an  ^12 

^21  ^22 
If  <f)  be  transformed  to  0'  by  the  substitutions 

^1=^11^1+612^2,      X2  =  b2iUi-\-b22U2y 

and  the  discriminant  d'  of  0'  be  formed,  then 

611  &12 

^21    ^22 

Thus  d  is  an  invariant  of  0. 

The  notation  for  the  w-ary  quadric  is 

in  which  dkk  is  the  coefficient  of  x^^,  while  that  of  XkXs  is  fl;^,. 
The  semi-derivatives  of  q  are,  upon  writing  a^s^ask, 

I   ?)q 
^*~2  "d^-^^^*^^"^^^*^^"^*  *  •■^^"»^"-     t^^^'  2, . . .  w] 

The  discriminant  is  therefore  the  symmetrical  determinant 
d=  ail  . . .  «ln 


«nl 


COMPOSITE   QUADRICS. 


47 


It  will  be  shown  in  Art.  42  that  the  discriminant  "of  every  quadric 
is  an  invariant.    This  also  follows  from  v Art.  30. 

Art.  36.    Composite  Quadrics. 

Whenever   a   quadric   is   resolvable   into  "V?     linear  factors 
its  discriminant  vanishes. 
To  prove  this,  let 

IIaksXkXs=(hXl  +  '  •  '+bnOCn)(ClXi+,  .  .  -\-CnXn). 

Equating  coefficients  gives 

L-o  —  Ij  2j  .  .  •  /*'— J 

Now,  the  product  of  the  two  rectangular  arrays 

hh  '  '  'bn     .     C1C2  .  .  .  Cn 
C1C2  .  .  .  Cn  hh  '  -  'hn 

is  the  aeterminant  (Art.  31) 

2hiCi  &2^i+6iC2    ,  ,  .hnCi+hiCn      =0 

61^2  +  ^2^1  262^2  •  •  •  bnC2+h2Cn 

hCn  +  hnC\       h2Cn  +  hnC2  -  •  -         2bnCn 

Hence,  substituting  from  the  equations  (i), 
d=    an  .  .  .  din    =0. 


ani  .  .  .  dr 


Art.  37.    Discriminant  of  Binary  Quantic  an  Invariant, 

When  a  binary  quantic  contains  a  square  factor  its  discrimi- 
nant vanishes. 

For,  any  such  quantic  is  of  the  form 

q=  (aiXi  +  a2X2f/(^u  yi), 

and  each  of  the  derivatives  -^ —  and  -^— contains  (ai;x:i+a2^2) 

as  a  factor.     Their  eliminant,  which  is  the  discriminant  of  q, 
must  therefore  vanish. 


43 


DETERMINANTS. 


A  square  factor  remains  such  after  linear  transformation. 
If,  therefore,  the  discriminant  vanishes,  that  of  the  transformed 
function  also  vanishes  and  thus  contains  the  first  as  a  factor. 
Hence,  the  discriminant  of  a  binary  quantic  is  an  invariant. 


Art.  38.    The  Jacobian. 

Let  >'i,  >'2>  •  •  •  yn  be  n  functions,  each  of  the  n  independent 
variables  ^i,  ^2, .  .  .  ^n-     Then  the  determinant 


J^ 


'byi 

^y\ 

dX2 


'dy2_ 

dX2 


'^yn 

dX2 


dyi       3j2  .  _  oyn 

dXn         dXn            'dXr? 

is  called  the  Jacobian  of  the  given  functions. 

The  notation 

J 

_d(yi,y2,- .  -yn) 

d(Xi,  X2,...  Xn) 

is  in  common  use,  being  suggested  by  the  close  analogy  between 
the  Jacqbian  and  the  ordinary  differential  coefficient. 
When  the  functions  are  linear,  thus: 

yi  =  auXi  +  a2iX2  + .  .  .  +  aniXn    [i=i,2, .  .  .n] 

it  follows  from  the  above  definition  that  the  Jacobian  is  the 
-determinant  of  the  coefficients.     That  is, 

/=   !  aiia22  ...  Ann  I  . 

When  the  functions  are  not  independent;  that  is,  when 
J^iyi)  y2y '  '  '  yn)=Of  the  Jacobian  vanishes. 

For,  differentiating  this  function  with  respect  to  each  of  the 
■variables  Xi,  X2,  .  .  .  Xn  gives  the  consistent  system 


dyi  dxi     'by  2  dxi 


JACOBIAN   OF   INDIRECT    FUNCTIONS. 


49 


and  eliminating  :~— , . . .  ^ —  from  these  equations  there  results 


'  dXn 


^J=o. 


Art.  39.    Jacobian  of  Indirect  Functions. 

When  )'i,  >'2>  •  •  •  Jn  are  each  functions  of  (^i,  ^21 . . .  (^m  these 
being  functions  each  of  x^,  ^2, . .  .  x^,  then 

d(x^  X2,...  Xn)        ^(Cl,  C2'  •  •  •  Cn)  '  ^(^U  ^2J  •  •  •  ^n)' 

This  may  be  demonstrated  by  writing  out  each  of  the  Jacobians 
in  the  second  member  in  determinant  form,  changing  columns 
into  rows  in  the  first,  multiplying  the  two  together  by  rows,  and 
interpreting  the  result  by  means  of  the  relation 

dyi_J^'^±^'^^^      ,  ^yi  '^Cn 

dXk      9^1  ^^k      3C2  ^^k      '  '  '       ^Cn  '^^k 

Thus,  for  w  =  2. 


d(yv  y^) .  ^(Cp  Q 
d{^vQ    d(x,,X2) 


^  ?a ,  ?ii  5C2 


dy,    'dy^ 
9Ci    9C2 

^Ci    9C2 


3G    ^ 
?Ci    ?C2 

9^2      9^2 


9,^1    9^2    'd^2    ^^3 


9^1    9:x;i     9(^2    ci^t^i 


?C2 
'dx^ 

d{yx,  y^ 
d(Xi,  x^y 


5^2  5Ci_^?>:_2 

90    9:^2    9^2 


9G 
9:x;- 


9^1    9;yi 
9^1    9^2 

dxj^    dx2 

It  may  be  shown  in  precisely  similar  manner  that,  when  the 
functions  yi,  y^,  . . .  yn  are  independent, 

^(yt>  yy  '  "  ^n)    d{x^,x.,.  .  .Xn)_ 
J(rVi,  :V2, .  .  .  Xr^  '  d{y\,  y.^,  .  .  .  >'„) 


50  DETERMINANTS. 

Art.  40.    The  Jacobian  a  Covariant. 

When,  in  the  preceding  article,  the  functions  ^^  C2>  •  •  •  C«  ^-re 
linear,  thus: 

(^i  =  a^iXi  +  a^iX^  + . . .  +  a^iXn ;    [i = i ,  2, . . .  w] 

then  (Art.  38) 

^fa> yi^'" yn) _ I ^  ^       ^^^|^(yi>>^2>»»-yn) 

d\X^'i  ^2>  •  •  •  -^n)  ^(Ci'  C2J  •  •  •  C«y 

Hence,  if  a  set  of  functions  be  subjected  to  linear  transformation, 
the  Jacobian  of  the  transformed  functions  is  equal  to  that  of  the 
given  functions  multiplied  by  the  modulus  of  the  transformation. 
That  is,  the  Jacobian  is  a  covariant  of  the  set  of  functions  from 
which  it  is  derived;  unless  these  functions  are  linear,  in  which 
case  it  is  an  invariant. 


Art.  41.    Jacobian  of  Implicit  Functions. 

When  the  functions  are  implicit,  thus: 

^i{yv  yv'  yn,  OC^,  X^,,..  Xn)  =  0\      [i=I,  2,  .  .  .  W] 

then 

d(yv  yv'"yn)_.       ^  ^  d{X,,   X^,  ...  Xn) 
d(Xi,  X^,...  Xn)  d((pij  02>  '  •  '  <t>n)    ' 

d(yv  yv'  yn) 

To  prove  this,  write  the  above  Jacobians  in  determinant  form,, 
change  columns  into  rows  in  the  first  member,  and  clear  of  frac- 
tions. This  gives,  representing  by  P  the  resulting  product  in 
the  first  member, 


P= 


dyi    dxk    ' ' '    'dyn   '^Xk 


THE    HESSIAN. 


51 


Now,  the  total  derivative  of  (j)i  with  respect  to  Xk  is 

dXk       '^Ji      '^OCk        ' ' '      '^Jn      ^Xk 

But,  since  x^^  ^2»  •  •  •  ^n  are  independent,  this  becomes 


The  above  product  thus  becomes 


P= 


9^- 


=  (-!)" 


?iXk 


=  (-l)n 


which  is  the  required  proof. 


Art.  42.    The  Hessian. 

The  Jacobian  of  the  first  differential  coefficients  of  a  function 
of  n  variables  is  called  the  Hessian  of  the  function.  Thus,  the 
Hessian  of  ^(x^,  x^^  .  .  .  x^^  is 


ff(<^)- 


Since 


/3<^     ^<i>          ^i>\ 

8V 

"^Xdx:     'dx,""'dXn} 

?iXn^X^ 

d{Xij      X2,     ...  Xn) 

dxfiXn 

dxk'^Xs    Zxgdx 

k 

the  Hessian  is  a  symmetric  determinant. 

The  Hessian  of  a  quadric  differs  from  the  discriminant  only 
by  a  numerical  factor. 

Let  the  function  ^  be  transformed  into  ^'  by  the  linear  sub- 


stitutions 


Xi=  a^iU^  +  ^21^2  + . . . -\- ani Xn.     [i=ii  2y  ,  .  .n] 


52  DETERMINANTS. 

Then  (Art.  40) 


But  ;z — -^ — =^ — p^,  and  the  above  equation  may  therefore  be 
written 

G^(2/p  ^2,  .  .  .  Un) 


=  a 


2 


CXn 


=  \a„n\'H  (<!>), 

That  is,  if  a  function  be  subjected  to  linear  transformation^ 
the  Hessian  of  the  transformed  function  will  equal  that  of  the 
given  function  multiplied  by  the  square  of  the  modulus  of  the 
transformation.  It  follows  that  the  Hessian  is  a  covariant  of  the 
function  from  which  it  is  derived;  unless  this  function  is  a  quadric, 
in  which  case  it  is  an  invariant  (Art.  35). 

Prob.  45.  Tell  whether  or  not  the  following  quadrics  are  prime: 
(i)  ^x^—gy^—$z^-\-i8yz+24ZX—i2xy; 

(2)  x^+z^+yz--2zx+xy;         (3)     gy^+i^yz—6zx-\-8xy. 

Prob.  46.  Find  the  values  of  X  in  order  that  each  of  the  following 
quadrics  may  be  composite: 

(i)  Xxy-\-szx+sy^+2Z^; 

(2)  2X^—sh^—'^^^^+^7yz+^zx—xy; 

(3)  /^x^+^z^-^yz+T,zx+2xy-^X{x^+Sy^—yz+szx+sxy), 
Prob.  47.  Find  the  Jacobian  of  the  functions 

yi  =  i-Xiy     ^'2=^1(1-^2),     >'3  =  ^1^2  (1-^3), 

...  yn  =  XiX2  .  .  .  Xn-i(l-Xn). 


THE    HESSIAN.  53 

Prob.  48.  Show,  1  y  means  of  their  Jacobian,  that  the  functions 

yi  =  {x-X2)(X2  +  X2),      y2={Xi-\rX2){X2-X3),      Ji,  =  ^^  (^'c  "  ^1 ) 

are  not  independent. 

d( X  "V^ 
Prob.  49.  Find        '      ;    having  given  x=pcosd  and  3'=^sin^,  in 

which  ^= aw,  d  =  hv. 

^    ,  ^ .        X  cos  tt  11  sin  X  -    ,     d(u,  v) 

Prob.  50.  Given =o,     — : —  =  o;     to  find      ,,       ,. 

V  cos  y  ^  sin  ^'  d{x,  y) 

Prob.  51.  Obtain  the  Hessian  and  the  discriminant  of:    (a)     the 

binary  cubic;   (b)  the  quaternary  quadric;   (c)  the  binary  quartic. 

Prob.  52.  Find  the  Hessian  of 

ax^  -\-by^-{-  cz^ + 2/yz  +  2gzx  +  2  hxy, 

and  also  that  of  the  same  function  transformed  by  the  substitutions. 

x=liX^-\-miyi  +  niZ%    y=l2x'+m2y^+n2Z%    z=lzx'-\-m2,y'-\-nzz'. 


INDEX. 


Alternant,  21. 
Arrays,  square,  10,  11. 
rectangular,  33. 

Binomial,  factors,  removal  of,  20. 

Cauchy,  7,  11,  27. 

Cauchy's  method  of  expansion,  26. 

Cayley,  7. 

Co-factors,  21,  37. 

Columns  and  rows,  10,  16,  17. 

Composition  of  parallel  lines,  19. 

Consistence  of  equations,  33. 

Covariants,  45,  50,  52. 

Cramer,  7. 

De  L'Hospital,  7. 
Development  of  determinants,  23. 
Differentiation,  28. 
Discriminant,  46,  47. 

Eliminant,  of  linear  systems,  35. 

Sylvester's  dialytic,  38. 
Equations,  linear  systems  of,  31,  40. 
Even  permutations,  8. 

Gauss,  7. 

Hessian,  51. 
Homogeneous  systems,  35. 

Implicit  functions,  Jacobian  of,  50. 
Indirect  functions,  Jacobian  of,  49. 


Interchange  of  elements,  9. 

of  rows  and  columns,  16. 
Invariant,  46,  47. 
Inversions  and  permanences,  8,  9. 

Jacobi,  7. 
Jacobian,  48. 

Lagrange,  7. 

Laplace,  34. 

Leibnitz,  7, 

Linear  systems  of  equations,  31,  33,  35. 

transformations,  40,  44. 
Lowering  the  order,  30. 

Matrix,  35. 
Minor,  21. 
Multiplication  of  determinants,  18,  40. 

of  matrices,  43. 

theorem,  40. 

Negative  permutations,  8,  10. 
Notation  of  determinants,  13. 

Odd  permutations,  8. 
Order  of  determinant,  12. 

Parallel  lines,  17. 

Permanences  and  inversions,  8,  9. 

Permutations,    positive    and    negative, 

8,  10. 
Polynomial  elements,  18. 
Positive  permutations,  8,  10. 


c.     c 
a,  c 


56 

Product  of  determinants,  40,  42. 
of  rectangular  arrays,  43. 

Quadric,  45,  51. 
Quantic,  45,  46,  47. 

Raising  the  order,  29. 
Reciprocal  determinants,  44. 
Resultant,  34. 


Sarrus's  rule,  14. 


INDEX 


Solution  of  linear  systems,  31. 

Sylvester,  7,  38. 

Sylvester's  method  of  elimination,  38. 

Transformations,  linear,  44. 

unimodular,  45. 

Unimodular  transformations,  45. 

Vandermonde,  7. 

Zero  determinants,  37. 
formulas,  25. 


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2 

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60 

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1 

SO 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,     i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,    5  00 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,    2  00 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,    2  00 

y^stwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo,     i   50 

"  "  "  "  Part  Two.     (Turnbull.) i2mo,    2  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests, 

Svo,  paper,        50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) Svo,    5  00 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,    i  50 

Poole's  Calorific  Power  of  Fuels Svo,    3  00 

Prescott  and  Winslow's  Elements  of  "Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,    i  25 

*  Reisig's  Guide  to  Piece-dyeing Svo,  25  00 

Richards  and  Woodman's  Air,  Water,  and   Food  from  a  Sanitary  Stand- 
point  Svo, 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo. 

Cost  of  Food,  a  Study  in  Dietaries i2mo, 

*  Richards  and  Williams's  The  Dietary  Computer Svo, 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) Svo,  morocco, 

Ricketts  and  Miller's  Notes  on  Assaying Svo, 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo, 

Disinfection  and  the  Preservation  of  Food Svo, 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory Svo, 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo, 

Ruddiman's  Incompatibilities  in  Prescriptions Svo, 

*  Whys  in  Pharmacy i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Omdorff.) Svo, 

Schimpf 's  Text-book  of  Volumetric  Analysis i2mo, 

Essentials  of  Volumetric  Analysis i2mo, 

*  Qualitative  Chemical  Analysis Svo, 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco. 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco, 

Stockbridge's  Rocks  and  Soils Svo, 

*  Tillman's  Elementary  Lessons  in  Heat Svo, 

*  Descriptive  General  Chemistry Svo, 

VTreadwell's  Qualitative  Analysis.     (Hall.) Svo, 

y        Quantitative  Analysis.     (Hall.) Svo, 

Turneaure  and  Russell's  Public  Water-supplies Svo, 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo, 

*  Walke's  Lectures  on  Explosives Svo, 

Ware's  Beet-sugar  Manufacture  and  Refining Small  Svo,  cloth, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo, 

Wassermann's  Immune  Sera :  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   i2mo. 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis Svo, 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo. 

Text-book  of  Chemical  Arithmetic i2mo, 

Whipple's  Microscopy  of  Drinking-water Svo, 

Wilson's  Cyanide  Processes i2mo, 

Chlorination  Process i2mo, 

Winton's  Microscopy  of  Vegetable  Foods Svo, 

WuUing's    Elementary    Course   in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo, 

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25 

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50 

CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS   OF   ENGINEERING 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo, 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches. 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo, 

Comstock's  Field  Astronomy  for  Engineers 8vo, 

Davis's  Elevation  and  Stadia  Tables 8vo, 

Elliott's  Engineering  for  Land  Drainage i2mo, 

Practical  Farm  Drainage i2mo, 

V*Fiebeger's  Treatise  on  Civil  Engineering Svo, 

Folwell's  Sewerage.     (Designing  and  Maintenance.) Svo, 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten Svo, 

French  and  I/es's  Stereotomy Svo, 

Goodhue's  Municipal  Improvements i2mo, 

Goodrich's  Economic  Disposal  of  Towns'  Refuse Svo, 

Gore's  Elements  of  Geodesy Svo, 

Hayford's  Text-book  of  Geodetic  Astronomy Svo, 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

Howe's  Retaining  Walls  for  Earth i2mo, 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  Svo, 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods Svo, 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscoit  and  Emory.) .  i2mo, 
Mahan's  Treatise  on  Civil  Engineering.     (1S73.)     (V/ood.). Svo, 

*  Descriptive  Geometry Svo, 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy Svo. 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morc^v. 

Nugent's  Plane  Surveying Svo, 

Ogden's  Sewer  Design i2mo, 

Patton's  Treatise  on  Civil  Engineering Svo  half  leather. 

Reed's  Topographical  Drawing  and  Sketching 4to, 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo, 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo, 
Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo, 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco. 

Wait's  Engineering  and  Architectural  Jurisprudence Svo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo, 

Sheep, 

Law  of  Contracts .• Svo, 

Warren's  Stereotomy — Problems  in  Stone-cutting Svo, 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco, 

Wilson's  Topographic  Surveying Svo, 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges . .  Svo,    2  00 

*       Thames  River  Bridge 4to,  paper,    5  00 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges Svo,    3  50 

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2 

00 

5 

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2 

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00 

3 

D'i' 

7 

00 

7 

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5 

00 

3 

50 

I 

50 

2 

50 

2 

00 

5 

00 

5 

00 

6 

00 

6 

50 

5 

00 

5 

50 

3 

00 

2 

50 

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25 

3 

50 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  . .  .8vo,  3  00 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  00 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  00 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  00 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses Svo,  i  25 

Bridge  Trusses Svo,  2  50 

Arches  in  Wood,  Iron,  and  Stone Svo,  2  50 

Howe's  Treatise  on  Arches Svo,  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel Svo,  2  00 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  00 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges : 

Part  I.     Stresses  in  Simple  Trusses Svo,  2  50 

Part  II.     Graphic  Statics Svo,  2  50 

Part  III.     Bridge  Design Svo,  2  50 

Part  IV.     Higher  Structures "Svo,  2  50 

Morison's  Memphis  Bridge 4to,  10  00 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  2  00 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume Svo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) .».  s Svo,  2  00 

Bovey's  Treatise  on  Hydraulics Svo,  5  00 

Church's  Mechanics  of  Engineering Svo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Hydraulic  Motors Svo,  2  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Folwell's  Water-supply  Engineering Svo,  4  00 

Frizell's  Water-power Svo,  5  00 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) Svo,  4  00 

Hazen's  Filtration  of  Public  Water-supply Svo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Water-works Svo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits _ Svo,  2  00 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

Svo,  4  00 

Merriman's  Treatise  on  Hydraulics Svo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics , Svo,  4  00 

Schuyler's  Reservoirs  for  Irrigation,  Water-power,  and  Domestic  Water- 
supply Large  Svo,  5  00 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  44c.  additional. ).4to,  6  00 

Turneaure  and  Russell's  Public  Water-supplies Svo,  5  00 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  00 

Water-supply  of  the  City  of  New  York  from  1658  to  1S95 4to,  10  00 

Williams  and  Hazen's  Hydraulic  Tables Svo,  i  50 

Wilson's  Irrigation  Engineering Small  Svo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 

Wood's  Turbines Svo,  2  50 

Elements  of  Analytical  Mechanics Svo,  3  00 

7 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo, 

Roads  and  Pavements 8vo, 

Black's  United  States  Public  Works Oblong  4to, 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering Svo, 

Byrne's  Highway  Construction Svo, 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo, 

Church's  Mechanics  of  Engineering Svo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to, 

♦Eckel's  Cements,  Limes,  and  Plasters Svo, 

Johnson's  Materials  of  Construction Large  Svo, 

Fowler's  Ordinary  Foundations Svo, 

*  Greene's  Structural  Mechanics , Svo, 

Keep's  Cast  Iron Svo, 

Lanza's  AppUed  Mechanics Svo, 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols Svo, 

Maurer's  Technical  Mechanics Svo, 

Merrill's  Stones  for  Building  and  Decoration Svo, 

Merriman's  Mechanics  of  Materials Svo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo, 

Patton's  Practical  Treatise  on  Foundations Svo," 

Richardson's  Modern  Asphalt  Pavements Svo, 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor., 

Rockwell's  Roads  and  Pavements  in  France i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Smith's  Materials  of  Machines i2mo. 

Snow's  Principal  Species  of  Wood Svo, 

Spalding's  Hydraulic  Cement i2mo. 

Text-book  on  Roads  and  Pavements i2mo, 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo, 

Thurston's  Materials  of  Engineering.     3  Parts Svo, 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy Svo, 

Part  II.     Iron  and  Steel Svo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo, 

Thurston's  Text-book  of  the  Materials  of  Construction Svo, 

Tillson's  Street  Pavements  and  Paving  Materials Svo, 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.) .  .  i6mo,  mor.. 

Specifications  for  Steel  Bridges i2mo. 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber .Svo, 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics Svo, 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  00 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brook's  Handbook  of  Street  Raihoad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables Svo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  00 

8 


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00 

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00 

5 

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so 

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50 

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3 

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50 

2 

50 

7 

50 

7 

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00 

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00 

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00 

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00 

5 

00 

3 

00 

4 

00 

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2S 

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00 

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00 

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00 

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00 

8 

00 

2 

00 

3 

50 

2 

50 

5 

00 

4 

00 

2 

00 

I 

25 

2 

00 

3 

00 

Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  00 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Raikoad  Engineers'  Field-book  and  Explores'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  I  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Nagle's  Field  Manual  for  Raikoad  Engineers i6mo,  morocco,  3  00 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork Svo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams Svo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

VWebb's  Railroad  Construction i6mo,  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  Svo,    5  00 


DRAWING. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bartlett's  Mechanical  Drawing Svo,  3  00 

*  "  "  "        Abridged  Ed Svo,  150 

Coolidge's  Manual  of  Drawing Svo,  paper  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines Svo,  4  00 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications Svo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective Svo,  2  oo^^, 

Jamison's  Elements  of  Mechanical  Drawing Svo,  2  5a 

Advanced  Mechanical  Drawing Svo,  2  00 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery Svo,  i  50 

Part  n.     Form,  Strength,  and  Proportions  of  Parts Svo,  3  00 

MacCord's  Elements  of  Descriptive  Geometry Svo,  3  oOv 

Kinematics;  or.  Practical  Mechanism Svo,  5  00 

Mechanical  Drawing ^ 4to,  4  00 

Velocity  Diagrams Svo,  i  50 

MacLeod's  Descriptive  Geometry Small  Svo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo,  i  50 

Industrial  Drawing.     (Thompson.) Svo,  3  50 

'O^oyer's  Descriptive  Geometry Svo,  2  00 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

2  00 


Reid's  Course  in  Mechanical  Drawing Svo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  00 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Schwamb  and  Merrill's  Element^  of  Mechanism Svo,  3  co 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.     (McMillan.) Svo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo,  3  00 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  i  00 

Drafting  Instruments  and  Operations i2mo,  i  23 

Manual  of  Elementary  Projection  Drawing i2mo,  i  50 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  i  00 

Plane  Problems  in  Elementary  Geometry i2mo,  i  2s 

9 


Warren's  Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  so- 
General  Problems  of  Shades  and  Shadows 8vo,  3  oa 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50. 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's    Kinematics  >nd    Power    of    Transmission.        (Hermann    and 

Klein.) 8vo,  5  Oq, 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo,  2  00 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  00 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  00 

ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo, 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  . .  .i2mo, 

Benjamin's  History  of  Electricity 8vo, 

Voltaic  Cell 8vo, 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo, 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco, 
Dolezalek's    Theory   of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo, 

Fkther's  Dynamometers,  and  the  Measurement  of  Power i2mo, 

filbert's  De  Magnete.     (Mottelay.) 8vo, 

Hanchett's  Alternating  Currents  Explained i2mo, 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

>5QHolman's  Precision  of  Measurements 8vo, 

Telescopic  Mirror-scale  Method,  Adjustments,  and  Tests.  . .  .Large  8vo, 
Xinzbrunner's  Testing  of  Continuous-current  Machines 8vo, 

■^^Landauer's  Spectrum  Analysis.     (Tingle.) 8vo, 

o^E^  Chateliers  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo. 
Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo, 

*  Lyons'3  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  Svo,  each, 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light Svo, 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo, 

^Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I Svo, 

Thurston's  Stationary  Steam-engines 1 Svo, 

*  Tillman|s  Elementary  Lessons  in  Heat Svo, 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo, 

Ulke's  Modern  Electrolytic  Copper  Refining Svo, 

LAW. 

*  Davis's  Elements  of  Law. Svo, 

*  Treatise  on  the  MiUtary  Law  of  United  States Svo, 

*  ^  Sheep, 

Manual  for  Courts-martial i6mo,  morocco, 

Wait's  Engineering  and  Architectural  Jurisprudence Svo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo 

Sheep, 

Law  of  Contracts 8vo, 

Winthrop's  Abridgment  of  Military  Law i2mo» 

10 


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MANUFACTURES. 

Bernadou's  Smokeless  Powder— Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  3  50 

Holland's  Iron  Founder ; i2mo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding ; i2mo,  3  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  i  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  00 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analsrsis  6f  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf' s  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  00 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  4  00 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  .  3  00 

Wood's  Rustless  Coating^s :  Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8 vo,  4  00 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

*  Bass's  Elements  of  Differential  Calculus i2mo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo,  i  00 

Compton's  Manual  of  Logarithmic  Computations i2mo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

y*  Dickson's  College  Algebra Large  i2mo,  i  50 

y^     Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  i  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Halsted's  Elements  of  Geometry 8vo,  i  75 

Elementary  Synthetic  Geometry 8vo,  i  50 

Rational  Geometry i2mo,  i  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size. paper,  15 

100  copies  for  5  00 

*  Mounted  on  heavy  cardboard,  8X 10  inches,  25 

10  copies  for  2  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,  3  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8 vo,  i  50 

u 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordiftates i2mo,     i  oo 

>4C2fohnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,     3  50 
"iCjolinson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,     i  50 

♦Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  00 

^<jLaplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.) .  i2mo,    2  00 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  00 

Trigonometry  and  Tables  published  separately Each,    2  00 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,    i  00 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     i  00 

No.  I.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bjuce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford^^JVeld.  "Vwo.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  \Bfo.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
byj  Mansfield  Merriman.  No.  11.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  00 

Merriman  and  Woodward's  Higher  Mathematics 8vo,    5  00 

^i^ierriman's  Method  of  Least  Squares 8vo,    2  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  00 
Differential  and  Integral  Calculus.     2  vols,  in  one Small  8vo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  00 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,    i  00 


MECHANICAL  ENGINEERING. 

MATERLALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

*  "  "  "        Abridged  Ed 8vo,  1  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  00 

"Carpenter's  Experimental  Engineering 8vo,  6  00 

Heating  and  Ventilating  Buildings 8vo,  4  00 

Cary's  Smoke  Suppression  in  Plants  using  Bittiminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  00 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6ma,  morocco,  2  50 

13 


Hutton's  The  Gas  Engine 8vo, 

Jamison's  Mechanical  Drawing 8vo, 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo, 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo, 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco, 

Kerr's  Power  and  Power  Transmission Svo, 

Leonard's  Machine  Shop,  Tools,  and  Methods Svo, 

*  Lorenz's  Modern  Refrigerating  Machinery.  (Pope,  Haven,  and  Dean.)  .  . Svo, 
MacCord's  Kinematics;  or,  Practical  Mechanism Svo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams Svo, 

MacFarland's  Standard  Reduction  Factors  for  Gases Svo, 

Mahan's  Industrial  Drawing.     (Thompson.) Svo, 

Poole's  Calorific  Power  of  Fuels Svo, 

Reid's  Course  in  Mechanical  Drawing Svo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo, 

Richard's  Compressed  Air i2mo, 

Robinson's  Principles  of  Mechanism Svo, 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo, 

Smith's  (O.)  Press-working  of  Metals Svo, 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo, 

Thurston's  Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill 
Work Svo, 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo, 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo, 

Weisbach's  Kinematics  and  the  Power  of  Transmission.  (Herrmann — 
Klein.) Svo, 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .Svo, 

Wolff's  Windmill  as  a  Prime  Mover Svo, 

Wood's  Turbines Svo, 


MATERULS  OP  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,    7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset Svo, 

Church's  Mechanics  of  Engineering Svo, 

*  Greene's  Structural  Mechanics Svo, 

Johnson's  Materials  of  Construction Svo, 

Keep's  Cast  Iron Svo, 

Lanza's  AppUed  Mechanics Svo, 

Martens's  Handbook  on  Testing  Materials.     (Henning.) Svo, 

Maurer's  Technical  Mechanics Svo, 

Merriman's  Mechanics  of  Materials Svo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Smith's  Materials  of  Machines i2mo, 

Thurston's  Materials  of  Engineering 3  vols.,  Svo, 

Part  II.     Iron  and  Steel Svo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo, 

Text-book  of  the  Materials  of  Construction Svo, 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber Svo, 

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Wood's  (De  V.)  Elements  of  Anal3rtical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  00 

STEAM-ENGINES  AND  BOILERS. 

B^rry's  Temperature-entropy  Diagram i2mo,  i  25 

>^arnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "  Engineering"  and  Electric  Traction  Pocket-book.  .  .   i6mo,  mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Goss's  Locomotive  Sparks Svo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  00 

Button's  Mechanical  Engineering  of  Power  Plants Svo,  s  00 

Heat  and  Heat-engines Svo,  5  00 

Kent's  Steam  boiler  Economy Svo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector. Svo,  i  50 

MacCord's  Slide-valves , Svo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors   Svo,  i  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines Svo,  5  00 

Valve-gears  for  Steam-engines Svo,  2  50 

Peabody  and  Miller's  Steam-boilers Svo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  Svo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives :  Simple   Compound,  and  Electric i2mo,  250 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) Svo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice Svo,  3  00 

Spangler's  Valve-gears Svo,  2  50 

Notes  on  Thermodynamics i2mo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Handy  Tables ' Svo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  Svo,  10  00 

Part  I.     History,  Structure,  and  Theory Svo,  6  00 

Part  II.     Design,  Construction,  and  Operation Svo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake Svo,  5  00 

Stationary  Steam-engines Svo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation Svo,  5  00 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) Svo,  5  00 

Whitham's  Steam-engine  Design Svo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  ..Svo,  4  00 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures   Svo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering Svo,  6  00 

Notes  and  Examples  in  Mechanics Svo,  2  00 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

14 


Cromwell's  Treatise  on  Toothed  Gearing i2mo,  I  SO 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics .' ,  . .  .8vo,  3  50 

Vol,    II.     Statics 8vo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

,.                                                        Vol.  II Small  4to,  10  00 

^vDurley's  Kinematics  of  Machines 8vo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  i  00 

>J^lather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  00 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  60 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

^^ohnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  00 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery - 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

.  Lanza's  Applied  Mechanics 8vo,  7  5° 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery,     (Pope,  Haven,  and  Dean.). 8vo,  4  00 
MacCord's  Kinematics;  or.  Practical  Mechanism 8vo,  5  00 

Velocity  Diagrams 8vo,  i  50 

^^ilaurer's  Technical  Mechanics 8vo,  4  00 

yllerriman's  Mechanics  of  Materials 8vo,  5  00 

yJT      Elements  of  Mechanics i2mo,  i  00 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  1 8vo,  2  so 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  00 

^Smith's  (O.)  Press-working  of  Metals 8vo,  3  00 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  00 

"^CtSmith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

•^^angler,  Greent,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  I  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.) .  8vo,  5  00 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

15 


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METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo, 

Vol,  II.     Gold  and  Mercury Svo, 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) i2mo, 

Keep's  Cast  Iron; Svo, 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe Svo, 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.  )i2mo. 

Metcalf's  Steel.     A  Manual  for  Steel-users.  .  .  . , i2mo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo, 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

Smith's  Materials  of  Machines i2mo, 

Thurston's  Materials  of  Engineering.     In  Three  Parts Svo, 

Part    II.     Iron  and  Steel Svo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo, 

Ulke's  Modern  Electrolytic  Copper  Refining Svo, 


MINERALOGY. 


Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco, 

Boyd's  Resources  of  Southwest  Virginia Svo, 

Map  of  Southwest  Virignia Pocket-book  form. 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) Svo, 

Chester's  Catalogue  of  Minerals Svo,  paper. 

Cloth, 

Dictionary  of  the  Names  of  Minerals Svo, 

Dana's  System  of  Mineralogy Large  Svo,  half  leather. 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  Svo, 

vText-book  of  Mineralogy Svo, 

Minerals  and  How  to  Study  Them i2mo, 

Catalogue  of  American  Localities  of  Minerals Large  Svo, 

Manual  of  Mineralogy  and  Petrography i2mo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo, 

V^akle's  Mineral  Tables Svo, 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo, 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small  Svo, 
Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses Svo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper,        50  \ 
Rosenbusch's   Microscopical  Physiography   of   the   Rock-making  Minerals. 

(Iddings.) Svo,    5  00    - 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo,    2  00 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form,  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  1  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor.,  25  00 

Eissler's  Modern  High  Explosives Svo,  4  00 

16 


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50 

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00 

50 

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00 

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25 

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50 

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4 

00 

2 

oo 

2 

50 

5 

oo 

2 

50 

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50 

2 

00 

4 

00 

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50 

I 

50 

2 

00 

I 

25 

Fowler's  Sewage  Works  Analyses i2mo, 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo, 

>/Ihlseng's  Manual  of  Mining 8vo, 

**  lles's  Lead-smelting.     (Postage  gc.  additional.) i2mo, 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo, 

O'DriscoU's  Notes  on  the  "freatment  of  Gold  Ores 8vo, 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

*  Walke's  Lectures  on  Explosives Svo, 

Wilson's  Cyanide  Processes i2mo, 

Chlorination  Process i2mo, 

Hydraulic  and  Placer  Mining i2mo, 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo, 


SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House i2mo, 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.)-  . 8vo, 

Water-supply  Engineering Svo, 

Fuertes's  Water  and  Public  Health i2mo. 

Water-filtration  Works i2mo, 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo, 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  Svo, 

Hazen's  Filtration  of  Public  Water-supplies Svo, 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Svo, 

Mason's  Water-supply.  ( Considered'principally  from  a  Sanitary  Standpoint)  Svo , 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo, 

Ogden's  Sewer  Design i2mo, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo, 

*  Price's  Handbook  on  Sanitation i2mo, 

Richards's  Cost  of  Food.     A  Study  in  bietaries i2mo, 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo, 

Richards  and  Woodman'-s  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  Svo, 

*  Richards  and  Williams's  The  Dietary  Computer Svo, 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage .8vo, 

Turneaure  and  Russell's  Public  Water-supplies Svo, 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo, 

Whipple's  Microscopy  of  Drinking-water Svo, 

Winton's  Microscopy  of  Vegetable  Foods Svo, 

Woodhull's  Notes  on  Military  Hygiene '. i6mo, 


MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.). . .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Svo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds Svo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1 824-1 894.. Small  Svo,  3  00 

Rostoski's.Serum  Diagnosis.     (Bolduan.) i2mo,  I  00 

Rotherham's  Emphasized  New  Testament , Large  Svo,  2  00 

17 


3 

00 

4 

00 

I 

50 

2 

50 

I 

00 

3 

50 

3 

00 

7  50 

4 

00 

I 

25 

2 

00 

I 

25 

J 

?o 

I 

00 

I 

00 

2 

00 

I 

50 

3 

50 

5 

00 

I 

00 

3 

so 

7 

50 

z 

so 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

Von  Behring's  Suppression  ot  Tuberculosis.     (Bolduan.). i2mo,  i  00 

Winslow's  Elements  of  Applied  Microscopy lamo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architectxire:  Plans  for  Small  Hospital.  1 2 mo,  i  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

18 


SHORT-TITLE     CATALOGUE 

OF  THE 

PUBLICATIONS 

OP 

JOHN   WILEY   &   SONS, 

New  York. 
LoHDOif:   CHAPMAN  &  HALL,  Limited, 


ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
at  net  prices  only,  a  double  asterisk  (**)  books  sold  under  the  rules  of  the  American 
Publishers'  Association  at  net  prices  subject  to  an  extra  charge  for  postage.  All  books 
are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding lamo,  Si  75 

Principles  of  Animal  Nutrition 8vo,  4  00 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement i2mo,  i  50 

Part  II.  Systematic  Pomology i2mo,  i  50 

Downing's  Fruits  and  Fruit-trees  of  America 8vo,  5  00 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo.  i  00 

Green's  Principles  of  American  Forestry i2mo,  i  50 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Well.) i2mo,  2  00 

Kemp's  Landscape  Gardening i2mo,  2  50 

Maynard's  Landscape  Gardening  as  Applied  to*[ome  Decoration i2mo,  i  50* 

^  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo,  i  50 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mOr  i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings i2mo.  2  50 

Bashore's  Sanitation  of  a  Country  House i2mo,  i  00 

Ber;s's  Buildings  and  Structures  of  American  Railroads 4to,  s  00 

Birkmire  s  Planning  and  Construction  of  American  Theatres 8vo,  3  00 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  00 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  so 

Skeleton  Construction  in  Buildings 8vo,  3  00 

Brigg's  Modern  American  School  Buildings Svo.  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings Svo,  4  00 

Freitag's  Architectural  Engineering Svo,  3  50 

Fireproofing  of  Steel  Buildings Svo,  2  50 

French  and  Ives's  Stereotomy. . , Svo,  2  50 

1 


I 

CO 

I 

50 

2 

SO 

75 

2 

oo 

5 

oo 

5 

oo 

4 

oo 

4 

oo 

5 

oc 

7 

5C 

4 

oo 

3 

oo 

1 

so 

3 

so 

2 

oo 

3 

oo 

6 

oo 

6 

so 

5 

oo 

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oo 

4 

oo 

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25 

I 

00 

Geriiaids  Guide  to  Saiiltey  XIo«se*inBpection i6mo, 

Theatre  Fires  and  Panics i2mo. 

♦Greene's  Structural  Mechanics  .  .  . 8vo, 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo, 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo, 

Kidder's  Architects'  and  Builders'  Pocket-book.  Rewritten  Edition.  i6mo,  mor., 

Merrill's  Stones  for  Building  and  Decoration Svo, 

Non-metallic  Minerals :   Their  Occurrence  and  Uses Svo, 

Monckton's  Stair-building ^ 4to, 

Patton's  Practical  Treatise  on  Foundations Svo, 

Peabody's  Naval  Architecture Svo, 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor., 

Sabin's  Industrial  and  Artistic  Technology  of  Paims  and  Varnish Svo, 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo, 

Snow's  Principal  Species  of  Wood Svo, 

Sondericker's  Grapnic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches, 

Svo, 

Towne's  Locks  and  Builders'  Hardware iSmo,  morocco, 

Wait's  Engineering  and  Architectural  Jurisprudence Svo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 

tectxire Svo, 

Sheep, 

Law  of  Contracts Svo, 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .Svo, 

Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 

Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo, 
The  World's  Columbian  Exposition  of  1S93 Large  4to, 


ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff's  Text-book  Ordnance  and  Guftnery Svo,  6  00 

Chase's  Screw  Propellers  and  Marine  Propulsion Svo,  3  00 

Cloke's  Gunner's  Examiner Svo,  i  50 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo.  3  00 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo,  7  00 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  00 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  00 

Durand's  Resistance  and  Propulsion  of  Ships Svo,  s  00 

*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  00 

Eissler's  Modern  High  Explosives Svo,  4  00 

*  Fiebeger's  Text-book  on  Field  Fortification Small  Svo,  2  00 

Hamilton's  The  Gunner's  Catechism iSmo,  i  00 

*  Hoff's  Elementary  Naval  Tactics Svo,  i  50 

lagalls's  Handbook  of  Problems  in  Direct  Fire Svo,  4  00 

*  Ballistic  Tables Svo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .Svo,  each,  6  00 

*  Mahan's  Permanent  Fortifications.    (Mercur.) Svo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

*  Mercur's  Attack  of  Fortified  Places i2mo  2  00 

*  Elements  of  the  Art  of  War Svo,  4  00 

2 


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